Number Theory

Problem 101

14. For any integer $a$ , show the following: (a) $\operatorname{gcd}(2 a+1,9 a+4)=1$ . (b) $\operatorname{gcd}(5 a+2,7 a+3)=1$ . (c) If $a$ is odd, then $\operatorname{gcd}(3 a, 3 a+2)=1$ .

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Problem 102

The method use to identiy the factors in the diagram below is called 6 12 24 2 2 3 2

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Problem 103

Which option shows the prime factorization of 200 in exponential form? $8 \cdot 25$ $2^{3} \cdot 5^{2}$ $2^{2} \cdot 5^{2}$ $2 \cdot 10^{2}$

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Problem 104

Exercice 1 1- Effectuer, en binaire, les opérations suivantes : a) $7 \mathrm{~B}_{(16)}+56,4(8)$ b) $\mathrm{A} 9_{(16)}-33_{(4)}$
2- Effectuer en octal : $726_{(8)}-535_{(8)}$ et en hexadécimal : $3 \mathrm{AD}_{(16)}+\mathrm{BOFF}_{(16)}$ . 3- Déterminer la base B sachant que $(25)_{\mathrm{B}}=(10111)_{2}$ 4- Convertir $94_{(10)}$ en binaire puis recopier et compléter le tableau suivant : \begin{tabular}{|c|c|c|c|c|c|c|c|c|} \hline Chiffres du nombre en base 2 & & & & & & & & \\ \hline Rang du chiffre & & & & & & & & \\ \hline Poids de chiffre & & & & & & & & \\ \hline Valeurs & & & & & & & & \\ \hline \end{tabular}
5- Déterminer x et y tel que : $(3 x 2, y 3)_{6}=(134,25)_{10}$ 6- Effectuer les conversions suivantes : a) $180_{(10)}=\mathrm{N}_{(2)}=\mathrm{N}_{(16)}$ b) $7 \mathrm{C}, \mathrm{B}_{(16)}=\mathrm{N}_{(8)}=\mathrm{N}_{(10)}$ c) $1010,011_{(2)}=\mathrm{N}_{(10)}=\mathrm{N}_{(8)}=\mathrm{N}_{(16)}$

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Problem 105

Convert the number BO in base 16 to base 2.
Answer:

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Problem 106

Find the highest common factor of the numbers 45, 120, and 180.

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Problem 107

Dimostrare che $\forall n \geq 0,11$ divide $9^{n+1}+2^{6 n+1}$

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Problem 108

Find the GCF of 35 and 80 using prime factors. What expression represents it? A. 5 B. 15 C. 8

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Problem 109

Find the prime factorization of 100. Options: A. $2 \times 2 \times 5 \times 5$ , B. $2 \times 5$ , C. $2 \times 2 \times 5$ , D. $10 \times 10$

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Problem 110

Choose a number to pair with 6 that is relatively prime: A. 6, B. 5, C. 12.

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Problem 111

Find the prime factorization of 24 using exponents for repeated factors. What is it?

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Problem 112

Find the prime factors of the number 105. What is its prime factorization? $105 =$

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Problem 113

Find the prime factorization of 76 using exponents for repeated factors. $76=$

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Problem 114

Find the greatest common divisor (gcd) of 16 and 50. What is the gcd? (Type a whole number.)

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Problem 115

Find the GCD of 99 and 66. The GCD of 99 and 66 is $$.

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Problem 116

Find the greatest common divisor of 105 and 175. What is it? (Type a whole number.)

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Problem 117

Determine if this statement is valid: "A prime number has three natural number factors." Choose A, B, or C.

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Problem 118

Find the prime factorization of 44 using exponents for repeated factors. $44 =$

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Problem 119

How many years in a teen's life are they a prime age? A. 3 B. 4 C. 1 D. 2

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Problem 120

Determine the number of significant figures in $20.00_{3}$ .

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Problem 121

Determine if this statement makes sense: "A prime number has three natural number factors." Explain your reasoning.

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Problem 122

Exercice 1 1- Effectuer, en binaire, les opérations suivantes: a) $7 \mathrm{~B}_{(16)}+56,4_{(8)}$ b) $\mathrm{A} 9_{(16)}-33_{(4)}$
2- Effectuer en octal : $726_{(8)}-535_{(8)}$ et en hexadécimal : $3 \mathrm{AD} 2_{(16)}+\mathrm{BOFF}_{(16)}$ . 3- Déterminer la base $B$ sachant que $(25)_{B}=(10111)_{2}$ 4- Convertir $94_{(10)}$ en binaire puis recopier et compléter le tableau suivant: \begin{tabular}{|l|l|l|l|l|l|l|l|l|} \hline \begin{tabular}{l} Chiffres du \\ nombre en base 2 \end{tabular} & & & & & & & & \\ \hline Rang du chiffre & & & & & & & & \\ \hline Poids de chiffre & & & & & & & & \\ \hline Valeurs & & & & & & & & \\ \hline \end{tabular}
5- Déterminer $x$ et $y$ tel que : $(3 x 2, y 3)_{6}=(134,25)_{10}$ 6- Effectuer les conversions suivantes : a) $180_{(10)}=\mathrm{N}_{(2)}=\mathrm{N}_{(16)}$ b) $7 \mathrm{C}, \mathrm{B}_{(16)}=\mathrm{N}_{(8)}=\mathrm{N}_{(10)}$ c) $1010,011_{(2)}=\mathrm{N}_{(10)}=\mathrm{N}_{(8)}=\mathrm{N}_{(16)}$
Exercice 2: 1) On considère la liste de matériels : Souris, routeur, carte mère, stylet, microprocesseur, écran plat, carte mémoire, câble RJ45, clé USB, modem, disque dur, graveur DVD, scanner, caméscope, imprimante, carte son, serveur, carte réseau, carte graphique. Relever dans cette liste: a) 3 Supports de stockage b) 3Composants internes à l'unité centrale c) 3 Equipements réseaux d) 3 Périphériques d'entrée e) 3 Cartes d'extension f) Un périphérique qui peut se connecter au port USB 2) Citer les différents bus système et donner le rôle de chacun d'entre eux

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Problem 123

Enter the prime factorization of 300 .
Use exponents for repeated factors. Be sure to use the * between different factors. For example, the prime factorization for 12 can be entered as $2 \wedge 2 * 3$ or $3 * 2 \wedge 2$ . $\square$ Submit Question

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Problem 124

Calcolare il resto della divisione di $5^{53}$ per 42. Soluzione. Calcolare il resto della divisione equivale a trova

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Problem 125

Find the prime factorization of 36, using exponents for repeated factors. What is it?

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Problem 126

Find the GCD of 42 and 105. The GCD of 42 and 105 is (Type a whole number.)

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Problem 127

5. If 21 divides $m$ , what else must divide $m$ ?

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Problem 128

11. True or false? Explain. a. If a counting number is divisible by 9 , it must be divisible by 3 . b. If a counting number is divisible by 3 and 11 , it must be divisible by 33 .

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Problem 129

13. Which of the following numbers are composite? Why? a. 12 b. 123 c. 1234 d. 12,345

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Problem 130

33. Fill in the blank. The sum of three consecutive counting numbers always has a divisor (other than 1 ) of $\qquad$ . Prove.

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Problem 131

11. True or false? Explain. a. If a counting number is divisible by 6 and 8 , it must be divisible by 48 . b. If a counting number is divisible by 4 , it must be divisible by 8 .

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Problem 132

16. If $n$ is a positive integer, the notation $n$ ! Which of the following is equal to a perfect square? (A) $\frac{(201)(191)}{1}$ (B) $\frac{(20!)(191)}{2}$ (C) $\frac{(20!)(19!)}{3}$ (D) $\frac{(20!)(19!)}{4}$ (E) $\frac{(20!)(191)}{5}$

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Problem 133

2. (6 marks)
Suppose that a store offers gift certificates in denominations of $\$ 25$ and $\$ 40$ . Make a conjecture regarding the possible total amounts you can form using these gift certificates, and prove it using a proof by strong induction. Follow the proof structure given in class and summarized in Lab 9.

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Problem 134

1 - 4444

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Problem 135

In 5-8, find the GCF for each pair of numbers.
5. 45,60
6. 24,100
7. 19, 22
8. 14,28

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Problem 136

In 13-16, find the LCM of each pair of numbers.
16. 4, 9
15. 3,4
13. 8,12
14. 6, 7

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Problem 137

17. Reasoning Mrs. James displayed the factor tree at the right. Complete the factor tree to find the number that has a prime factorization of $2^{4} \times 3$ .

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Problem 138

The sum of 132 and 402 is exactly divisible by which number?
F 4 G 6 H 8 J 9 If None of these

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Problem 139

Is the number 15 prime, composite, or neither?

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Problem 140

Find the prime factors of 25.

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Problem 141

Find the prime factors of 245.

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Problem 142

Find the highest common factor (HCF) of $A=2 \times 3^{4} \times 5$ and $B=2^{3} \times 3^{2} \times 5^{2}$ . HCF = 12.

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Problem 143

What is the prime factorization of each expression?
$\text { A) } \begin{aligned} 8 x y & =8 \cdot x \cdot y \\ 12 x & =12 \cdot x \end{aligned}$ B) $\begin{array}{l} 8 x y=1 \cdot 8 \cdot x \cdot y \\ 12 x=1 \cdot 12 \cdot x \end{array}$ C) $\begin{array}{l} 8 x y=2 \cdot 4 \cdot x \cdot y \\ 12 x=2 \cdot 6 \cdot x \end{array}$
$\text { D) } \begin{aligned} 8 x y & =2 \cdot 2 \cdot 2 \cdot x \cdot y \\ 12 x & =2 \cdot 2 \cdot 3 \cdot x \end{aligned}$

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Problem 144

All previous units in the textbook have used the "Base-10" or "Decimal" numbering system. Base-10 is the most common of all numbering systems in use today, worldwide, but it is not the only one in use. Computers operate on Binary calculations. Therefore number systems of Base-8 (octal) and Base-16 (hexadecimal) are useful to computers, whereas Base-10 cannot be readily used by computers without an interpretation algorhythm, which slows everything down.
When a computer operates in the Octal system, the digit word length is three data bits long. When a computer operates in the hexadecimal system, the digit word length is only four data bits long. Please see the table on the page 251 of your textbook. (The table on page 248 has errors in the binary column)
Don't be confused by the hexadecimal system use of the letters A B C D E and F as digits. Please note that digit $D$ is equal in value to, but does not represent decimal number 13 . It is a distinct digit in the "hex" system. It works because it is a single digit, where 13 would not work because 13 is two digits.
Use your textbook for guidance where needed. Feel free to "Google" these questions. Question: What is the Binary equivalent to the "Hex" number 3F? (A) 111111 (B) 11110 (C) 111011 (D) 100111

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Problem 145

Find the prime factors of 338.

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Problem 146

Find the prime factorization of 72 and the largest perfect square factor.

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Problem 147

Find the smallest number of bingo chips that leaves a remainder of 2 when divided by 9, 10, or 12. Options: 80, 182, 360, 542, 1080.

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Problem 148

$\text{Ерөнхий шинж чанартай, ялгаатай юмсын цуглуулгыг олонлог гэнэ. Тэгвэл дараах зүйлүүдээс аль нь зохиомол тоон олонлогт багтах вэ?}$ $18, 72, 81, 13, 7, 64, -7, 17$ $\text{зохиомол тоо анхны тоо гэж юу вэ}$ $\text{аль нь зохиомол тоо вэ}$

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Problem 149

4. Assuming that $\operatorname{gcd}(a, b)=1$ , prove the following: (b) $\operatorname{gcd}(2 a+b, a+2 b)=1$ or 3 .

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Problem 150

Number Theory and Fractions Unit Test
Express the prime factorization of 1,200 as a product of prime numbers with exponents. (2 points)
4. $\square$ $\square$ ${ }^{2}=1,200$ $\square$

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Problem 151

1 \&page=18idWebuser=5901475\&idSection =19532928self= True\& menu
UNIT 7 LESSON 14 Number Theory and Fractions Number Theory and Fractions Mark as Comp COURSE OUTLINE
Number Theory and Fractions Unit Test COURSE TOOLS
A track team needed to run or jog 20 laps for their practice. Of the total 20 laps, Sara completed $\frac{4}{5}$ , Jamie completed $\frac{5}{8}$ , Keenan completed $\frac{5}{9}$ , Maya completed 0.833 , and Trey completed 0.875 . Identify the descending numeric order of the laps completed. Be sure to input the original form of each number. (2 points) Item 8 Item 9 Item 10 ftem 11 liem 12 Item 13 Rem 14

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Problem 152

Consider the numbers 13, 16, 27, and 41. a. Which of these numbers are prime? How do you know? b. Which of these numbers are composite? How do you know? (2 points)

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Problem 153

Find the 4-digit ATM code for Mr. Nolan using the prime factors of 84 in increasing order.

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Problem 154

Find the 4-digit ATM code for Mr. Nolan using the prime factors of 84 listed in order.

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Problem 155

List the first 5 multiples, and find ALL the factors of 21.
Multiples:
Factors: Prime or Composite?

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Problem 156

4. Assuming that $\operatorname{gcd}(a, b)=1$ , prove the following: (b) $\operatorname{gcd}(2 a+b, a+2 b)=1$ or 3 .

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Problem 157

Identify each number as prime or composite. Then list all the factors. 5) 2 6) 37 7) 27 8) 54 9) 19 10) 63

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Problem 158

1) 2) 3)
Prime Factors ${ }_{-} x_{-} x_{-} x_{-}=60$ Prime Factors Prime Factors ${ }_{-} x_{-} x_{-}=99$ $-x_{-} x_{-} x_{-} x_{-}=48$

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Problem 159

which of the fallowing Pairs has GCf of 1 ? (A) $\begin{array}{l} 8 \text { and } 12 \\ 9 \text { and } 28 \\ 10 \text { and } 15 \\ 14 \text { and } 21 \end{array}$

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Problem 160

The building was built in the year $1865$ .

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Problem 161

What is the minimum number of powers of 2 prizes that can total \$1020?

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Problem 162

What is the minimum number of powers of 2 prizes that can sum to $\$ 818$ ?

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Problem 163

Find the prime factorization of these numbers: a) 882 b) 6,760.

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Problem 164

For the pair of numbers, find a third whole number such that the three numbers form a Pythagorean triple. $40, 42$ The third number of the Pythagorean triple is $\boxed{\phantom{000}}$ . (Type a whole number.)

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Problem 165

Find the units digit of $3^{40}$ . Options: (A) 1 (B) 3 (C) 6 (D) 7 (E) 9

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Problem 166

Find two prime numbers that multiply to equal 55. $p_1 \times p_2 = 55$

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Problem 167

Find factor pairs of 27 and circle the prime numbers among them.

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Problem 168

What is the minimum number of prizes, all powers of 2, that can total \$820?

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Problem 169

Mrs. Stevenson has 24 students. She warts to use her knowiedge of mul ples to pur they students into equal groups. Which is not a possible number of groups lilirs Steversan cars have if she wants her 24 students in equal groups? A. 3 groups B. 4 groups C. 5 groups D. 6 graps

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Problem 170

ex 3 and 5 3: 3, 6, 9, 12, $15$ , 18 5: 5, 10, $15$ , 20 The LCM is 15. a 4: 6: LCM: b

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Problem 171

b $3$ $7$ LCM:

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Problem 172

$xy$ عددان زوجيان $\Leftarrow xy$ عدد زوجي
إختر واحداً: صح خطأ
سؤال 2 غير مجاب عليه بعد الدرجة من 1.00 علم هذا السؤال
تمثل الأعداد بالنظام الثنائي بوساطة قوى الأساس 2 .a 8 .b 10 .c 16 .d

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Problem 173

Sparx Maths New! Multi Part Question - when you answer this question we'll mark each part individually Bookwork code: 1B Calculator not allowed
For each number, decide whether it is prime or not prime: a) 51 b) 87

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Problem 174

d) Phuntsho factored every multiple of 4 from 4 to 40 into prime factors. He used a matrix to show how many times each prime factor (2,3,5 and 7) appeared in each number. Create Phuntsho's matrix. [2]

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Problem 175

Find the prime factors of the number 27 using a tree diagram.

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Problem 176

7. Add or subtract in the indicated base.
$101010_2$ $- 111011_2$

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Problem 177

Question three: " 5 points"
1. Solve the following linear congruence $140x \equiv 56 \pmod{252}$
2. How many incongruent solutions this equation have?

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Problem 178

b. $\pi$ è un numero irrazionale.
Questo significa che: (A) è un numero decimale periodico semplice (B) è un numero decimale limitato
C è un numero decimale periodico misto (D) è un numero decimale illimitato non periodico

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Problem 179

16. Convert to the indicated base. a) $1101_{2}$ To base 5 b) 243 s To base 2

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Problem 180

Convert $86_{\text {twelve }}$ to decimal. What is the decimal value?

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Problem 181

Convert the number $86_{\text{twelve}}$ to decimal.

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Problem 182

Find the GCF of 180, 270, and 360. What is $\operatorname{GCF}(180,270,360)$ ?

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Problem 183

Factor the number 96 and list its prime factors in order.

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Problem 184

Anthony goes to the library every $4^{\text {th }}$ day, and Giovani goes to the library every $6^{\text {th }}$ day. When will they go to the library on the same day?

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Problem 185

7/10
Which expression shows the prime factorization of $110 ?$ $2 \cdot 5 \cdot 11$ $100+10$ $5 \cdot 22$ $10^{\wedge} 2$

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Problem 186

Question (3): 7 points) Show that an integer x is divisible by 4 if and only if the last two digits is divisible by 4 .
Is th integer 42412311145633677880000000042 is divisible by 4 ?

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Problem 187

1-1 List the binary, octal, and hexadecimal numbers from 16 to 31. 1-2 What is the exact number of bits in a system that contains (a) 32 K bits; (b) 50 K bits; (c) 32 M bits?
1-3 What is the largest binary number that can be obtained with 16 bits? What is its decimal equivalent? 1-4 Give the binary value of a 24-bit number with the hexademical equivalent of F3A7C2. What is the octal equivalent of the binary number? 1-5 Convert the following binary numbers to decimal: $101110,1110101.11$ , and 110110100 . 1-6 Convert the following decimal numbers to binary: 1231, 673, 104 , and 1998. 1-7 Convert the following numbers with the indicated bases to decimal: $(12121)_{3},(4310)_{5}$ , and (198) ${ }_{12}$ . 1-8 Convert the following numbers from the given base to the other three bases listed in the table. \begin{tabular}{cccc} $\underline{\text { Decimal }}$ & & Binary & $\underline{\text { Octal }}$ \\ 225.225 & $?$ & & \multirow{2}{\text{Hexadecimal}}{} \\ $?$ & 11010111.11 & $?$ & $?$ \\ $?$ & $?$ & 623.77 & $?$ \\ $?$ & $?$ & $?$ & 2AC5.D \end{tabular}
1-9 Add and multiply the following numbers without converting to decimal. (a) $(367)_{8}$ and (715) 8 (b) $(15 \mathrm{~F})_{16}$ and (A7) ${ }_{16}$ (c) $(110110)_{2}$ and $(110101)_{2}$
1-10 Convert the following decimal numbers to the indicated bases using the methods of Examples 1-3 and 1-6. (a) 7562.45 to octal (b) 1938.257 to hexadecimal (c) 175.175 to binary.
1-11 Perform the following division in binary: $11111111 \div 101$ . 1-12 Obtain the 9's complement of the following 8-digit decimal numbers: 12349876, 00980100, 90009951, and 00000000 . 1-13 Obtain the 10's complement of the following 6-digit decimal numbers: 123900, 090657, 100000, and 000000. 1-14 Obtain the 1's and 2's complements of the following binary numbers: 10101110, 10000001, 10000000,00000001 , and 00000000 .

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Problem 188

14. Sia $K=31 \cdot 32 \cdot 33 \cdots \cdots 50 \cdot 51$ il prodotto dei numeri interi da 31 a 51 . Quanti sono i numeri primi il cui quadrato divide $K$ ? ((tenere presente che il numero 1 non è primo)) (A) 7 (B) 6 (C) 10 (D) 8 (E) 5

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Problem 189

Identify the greatest common factor of $36 z$ and 12 .
Answer Attempt 1 out of 2

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Problem 190

17. Add or subtract in the indicated base. \begin{tabular}{ccc} $101010_{2}$ & 210 & \\ $+111011_{2}$ & $3=11$ & $111111_{2}$ \\ \hline $19110_{2}$ & & $+111000_{2}$ \\ \hline $110111_{2}$ \end{tabular} $\begin{array}{l} \begin{array}{lll} 3424_{5} & & \\ 2414_{5} & 6 & =10 \\ 13435 & 7 & =12 \\ & 8 & 11 \\ & 9 & 1344_{5} \\ & & \\ & & 1033_{5} \\ 2432 \end{array} \end{array}$

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Problem 191

Bobby is serving vegetables at a soup kitchen. He has 36 carrot sticks and 44 baby potatoes that he wants to divide evenly, with no food left over. What is the greatest number of plates Bobby can prepare? $\square$ plates

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Problem 192

What is the highest common factor (HCF) of 15 and 3 ?

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Problem 193

Complete the following. (a) Find the greatest common factor (GCF) of 21 and 12. $\text { GCF }=$

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Problem 194

which has products that are the Shade the yellow shaded numbers in each row.
2. What pattern do you see? \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline $x$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 1 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 2 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\ \hline 3 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 27 \\ \hline 4 & 4 & 8 & 12 & 16 & 20 & 24 & 28 & 32 & 36 \\ \hline 5 & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 \\ \hline 6 & 6 & 12 & 18 & 24 & 30 & 36 & 42 & 48 & 54 \\ \hline 7 & 7 & 14 & 21 & 28 & 35 & 42 & 49 & 56 & 63 \\ \hline 8 & 8 & 16 & 24 & 32 & 40 & 48 & 56 & 64 & 72 \\ \hline 9 & 9 & 18 & 27 & 36 & 45 & 54 & 63 & 72 & 81 \\ \hline \end{tabular}
3. Explain why this pattern is true.

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Problem 195

Write 180 as a product of its prime factors: $180 = 2^2 \times 3^2 \times 5$ .

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Problem 196

Find the prime factorization of 2 and list the factors in ascending order (e.g., $2 \times 2 \times 3 \times 5$ ).

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Problem 197

Find the prime factorization of 20: $2 \times 2 \times 5$ .

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Problem 198

Find the prime factorization of 8, listing factors from least to greatest, e.g., $2 \times 2 \times 2$ .

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Problem 199

Find the prime factorization of 4, listing the factors from least to greatest (e.g., $2 \times 2$ ).

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Problem 200

The remainder obtained when $17^{201}$ is divided by 18 , is :

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1 2 3

Number Theory

Problem 101

14. For any integer aaa, show the following: (a) gcd⁡(2a+1,9a+4)=1\operatorname{gcd}(2 a+1,9 a+4)=1gcd(2a+1,9a+4)=1. (b) gcd⁡(5a+2,7a+3)=1\operatorname{gcd}(5 a+2,7 a+3)=1gcd(5a+2,7a+3)=1. (c) If aaa is odd, then gcd⁡(3a,3a+2)=1\operatorname{gcd}(3 a, 3 a+2)=1gcd(3a,3a+2)=1.

Problem 102

The method use to identiy the factors in the diagram below is called 6 12 24 2 2 3 2

Problem 103

Which option shows the prime factorization of 200 in exponential form? 8⋅258 \cdot 258⋅25 23⋅522^{3} \cdot 5^{2}23⋅52 22⋅522^{2} \cdot 5^{2}22⋅52 2⋅1022 \cdot 10^{2}2⋅102

Problem 104

Problem 105

Convert the number BO in base 16 to base 2.Answer:

Problem 106

Find the highest common factor of the numbers 45, 120, and 180.

Problem 107

Dimostrare che ∀n≥0,11\forall n \geq 0,11∀n≥0,11 divide 9n+1+26n+19^{n+1}+2^{6 n+1}9n+1+26n+1

Problem 108

Find the GCF of 35 and 80 using prime factors. What expression represents it? A. 5 B. 15 C. 8

Problem 109

Find the prime factorization of 100. Options: A. 2×2×5×52 \times 2 \times 5 \times 52×2×5×5, B. 2×52 \times 52×5, C. 2×2×52 \times 2 \times 52×2×5, D. 10×1010 \times 1010×10

Problem 110

Choose a number to pair with 6 that is relatively prime: A. 6, B. 5, C. 12.

Problem 111

Find the prime factorization of 24 using exponents for repeated factors. What is it?

Problem 112

Find the prime factors of the number 105. What is its prime factorization? 105=105 = 105=

Problem 113

Find the prime factorization of 76 using exponents for repeated factors. 76= 76= 76=

Problem 114

Find the greatest common divisor (gcd) of 16 and 50. What is the gcd? (Type a whole number.)

Problem 115

Find the GCD of 99 and 66. The GCD of 99 and 66 is $$.

Problem 116

Find the greatest common divisor of 105 and 175. What is it? (Type a whole number.)

Problem 117

Determine if this statement is valid: "A prime number has three natural number factors." Choose A, B, or C.

Problem 118

Find the prime factorization of 44 using exponents for repeated factors. 44= 44 = 44=

Problem 119

How many years in a teen's life are they a prime age? A. 3 B. 4 C. 1 D. 2

Problem 120

Determine the number of significant figures in 20.00320.00_{3}20.003​.

Problem 121

Determine if this statement makes sense: "A prime number has three natural number factors." Explain your reasoning.

Problem 122

Problem 123

Enter the prime factorization of 300 .Use exponents for repeated factors. Be sure to use the * between different factors. For example, the prime factorization for 12 can be entered as 2∧2∗32 \wedge 2 * 32∧2∗3 or 3∗2∧23 * 2 \wedge 23∗2∧2. □\square□ Submit Question

Problem 124

Calcolare il resto della divisione di 5535^{53}553 per 42. Soluzione. Calcolare il resto della divisione equivale a trova

Problem 125

Find the prime factorization of 36, using exponents for repeated factors. What is it?

Problem 126

Find the GCD of 42 and 105. The GCD of 42 and 105 is (Type a whole number.)

Problem 127

5. If 21 divides mmm, what else must divide mmm ?

Problem 128

11. True or false? Explain. a. If a counting number is divisible by 9 , it must be divisible by 3 . b. If a counting number is divisible by 3 and 11 , it must be divisible by 33 .

Problem 129

13. Which of the following numbers are composite? Why? a. 12 b. 123 c. 1234 d. 12,345

Problem 130

33. Fill in the blank. The sum of three consecutive counting numbers always has a divisor (other than 1 ) of \qquad . Prove.

Problem 131

11. True or false? Explain. a. If a counting number is divisible by 6 and 8 , it must be divisible by 48 . b. If a counting number is divisible by 4 , it must be divisible by 8 .

Problem 132

Problem 133

Problem 134

1 - 4444

Problem 135

In 5-8, find the GCF for each pair of numbers.5. 45,606. 24,1007. 19, 228. 14,28

Problem 136

In 13-16, find the LCM of each pair of numbers.16. 4, 915. 3,413. 8,1214. 6, 7

Problem 137

17. Reasoning Mrs. James displayed the factor tree at the right. Complete the factor tree to find the number that has a prime factorization of 24×32^{4} \times 324×3.

Problem 138

The sum of 132 and 402 is exactly divisible by which number?F 4 G 6 H 8 J 9 If None of these

Problem 139

Is the number 15 prime, composite, or neither?

Problem 140

Find the prime factors of 25.

Problem 141

Find the prime factors of 245.

Problem 142

14. For any integer $a$ , show the following: (a) $\operatorname{gcd}(2 a+1,9 a+4)=1$ . (b) $\operatorname{gcd}(5 a+2,7 a+3)=1$ . (c) If $a$ is odd, then $\operatorname{gcd}(3 a, 3 a+2)=1$ .

Which option shows the prime factorization of 200 in exponential form? $8 \cdot 25$ $2^{3} \cdot 5^{2}$ $2^{2} \cdot 5^{2}$ $2 \cdot 10^{2}$

Convert the number BO in base 16 to base 2.
Answer:

Dimostrare che $\forall n \geq 0,11$ divide $9^{n+1}+2^{6 n+1}$

Find the prime factorization of 100. Options: A. $2 \times 2 \times 5 \times 5$ , B. $2 \times 5$ , C. $2 \times 2 \times 5$ , D. $10 \times 10$

Find the prime factors of the number 105. What is its prime factorization? $105 =$

Find the prime factorization of 76 using exponents for repeated factors. $76=$

Find the prime factorization of 44 using exponents for repeated factors. $44 =$

Determine the number of significant figures in $20.00_{3}$ .

Enter the prime factorization of 300 .
Use exponents for repeated factors. Be sure to use the * between different factors. For example, the prime factorization for 12 can be entered as $2 \wedge 2 * 3$ or $3 * 2 \wedge 2$ . $\square$ Submit Question

Calcolare il resto della divisione di $5^{53}$ per 42. Soluzione. Calcolare il resto della divisione equivale a trova

5. If 21 divides $m$ , what else must divide $m$ ?

33. Fill in the blank. The sum of three consecutive counting numbers always has a divisor (other than 1 ) of $\qquad$ . Prove.

In 5-8, find the GCF for each pair of numbers.
5. 45,60
6. 24,100
7. 19, 22
8. 14,28

In 13-16, find the LCM of each pair of numbers.
16. 4, 9
15. 3,4
13. 8,12
14. 6, 7

17. Reasoning Mrs. James displayed the factor tree at the right. Complete the factor tree to find the number that has a prime factorization of $2^{4} \times 3$ .

The sum of 132 and 402 is exactly divisible by which number?
F 4 G 6 H 8 J 9 If None of these

Find the highest common factor (HCF) of $A=2 \times 3^{4} \times 5$ and $B=2^{3} \times 3^{2} \times 5^{2}$ . HCF = 12.

4. Assuming that $\operatorname{gcd}(a, b)=1$ , prove the following: (b) $\operatorname{gcd}(2 a+b, a+2 b)=1$ or 3 .

Number Theory and Fractions Unit Test
Express the prime factorization of 1,200 as a product of prime numbers with exponents. (2 points)
4. $\square$ $\square$ ${ }^{2}=1,200$ $\square$

List the first 5 multiples, and find ALL the factors of 21.
Multiples:
Factors: Prime or Composite?

4. Assuming that $\operatorname{gcd}(a, b)=1$ , prove the following: (b) $\operatorname{gcd}(2 a+b, a+2 b)=1$ or 3 .

1) 2) 3)
Prime Factors ${ }_{-} x_{-} x_{-} x_{-}=60$ Prime Factors Prime Factors ${ }_{-} x_{-} x_{-}=99$ $-x_{-} x_{-} x_{-} x_{-}=48$

which of the fallowing Pairs has GCf of 1 ? (A) $\begin{array}{l} 8 \text { and } 12 \\ 9 \text { and } 28 \\ 10 \text { and } 15 \\ 14 \text { and } 21 \end{array}$

The building was built in the year $1865$ .

What is the minimum number of powers of 2 prizes that can sum to $\$ 818$ ?

For the pair of numbers, find a third whole number such that the three numbers form a Pythagorean triple. $40, 42$ The third number of the Pythagorean triple is $\boxed{\phantom{000}}$ . (Type a whole number.)

Find the units digit of $3^{40}$ . Options: (A) 1 (B) 3 (C) 6 (D) 7 (E) 9

Find two prime numbers that multiply to equal 55. $p_1 \times p_2 = 55$

ex 3 and 5 3: 3, 6, 9, 12, $15$ , 18 5: 5, 10, $15$ , 20 The LCM is 15. a 4: 6: LCM: b

b $3$ $7$ LCM:

Sparx Maths New! Multi Part Question - when you answer this question we'll mark each part individually Bookwork code: 1B Calculator not allowed
For each number, decide whether it is prime or not prime: a) 51 b) 87