Math / Numbers & Operations

QuestionFind the prime factors of the following numbers and express them in the index form : 4.95 2.230
3. 105 8.110

Studdy Solution

STEP 1

1. Prime factorization involves breaking down a number into its prime factors.
2. The prime factors of a number are the prime numbers that divide the number exactly, without leaving a remainder.
3. Index form (or exponential form) involves expressing repeated prime factors using exponents.

STEP 2

1. Prime factorize the number 95.
2. Prime factorize the number 230.
3. Prime factorize the number 105.
4. Prime factorize the number 110.

STEP 3

Determine if 95 is divisible by the smallest prime number, which is 2. Since 95 is odd, it is not divisible by 2.

STEP 4

Check the next smallest prime number: 3. The sum of the digits of 95 is $9+5=14$ , which is not divisible by 3. Hence, 95 is not divisible by 3.

STEP 5

Check the next smallest prime number: 5. Since 95 ends in 5, it is divisible by 5. $95 \div 5 = 19$

STEP 6

Determine if 19 is a prime number. Since 19 is not divisible by 2, 3, or 5 (the prime numbers less than its square root), 19 is a prime number.

STEP 7

Express the prime factors of 95 in index form: $95 = 5^1 \times 19^1$

STEP 8

Determine if 230 is divisible by the smallest prime number, which is 2. Since 230 is even, it is divisible by 2. $230 \div 2 = 115$

STEP 9

Check if 115 is divisible by the next smallest prime number: 3. The sum of the digits of 115 is $1+1+5=7$ , which is not divisible by 3. Hence, 115 is not divisible by 3.

STEP 10

Check the next smallest prime number: 5. Since 115 ends in 5, it is divisible by 5. $115 \div 5 = 23$

STEP 11

Determine if 23 is a prime number. Since 23 is not divisible by 2, 3, or 5 (the prime numbers less than its square root), 23 is a prime number.

STEP 12

Express the prime factors of 230 in index form: $230 = 2^1 \times 5^1 \times 23^1$

STEP 13

Determine if 105 is divisible by the smallest prime number, which is 2. Since 105 is odd, it is not divisible by 2.

STEP 14

Check the next smallest prime number: 3. The sum of the digits of 105 is $1+0+5=6$ , which is divisible by 3. Hence, 105 is divisible by 3. $105 \div 3 = 35$

STEP 15

Check if 35 is divisible by the next smallest prime number: 5. Since 35 ends in 5, it is divisible by 5. $35 \div 5 = 7$

STEP 16

Determine if 7 is a prime number. Since 7 is not divisible by 2 or 3 (the prime numbers less than its square root), 7 is a prime number.

STEP 17

Express the prime factors of 105 in index form: $105 = 3^1 \times 5^1 \times 7^1$

STEP 18

Determine if 110 is divisible by the smallest prime number, which is 2. Since 110 is even, it is divisible by 2. $110 \div 2 = 55$

STEP 19

Check if 55 is divisible by the next smallest prime number: 3. The sum of the digits of 55 is $5+5=10$ , which is not divisible by 3. Hence, 55 is not divisible by 3.

STEP 20

Check the next smallest prime number: 5. Since 55 ends in 5, it is divisible by 5. $55 \div 5 = 11$

STEP 21

Determine if 11 is a prime number. Since 11 is not divisible by 2, 3, or 5 (the prime numbers less than its square root), 11 is a prime number.

STEP 22

Express the prime factors of 110 in index form: $110 = 2^1 \times 5^1 \times 11^1$
Solution: The prime factorization of the given numbers in index form is as follows: $95 = 5^1 \times 19^1$ $230 = 2^1 \times 5^1 \times 23^1$ $105 = 3^1 \times 5^1 \times 7^1$ $110 = 2^1 \times 5^1 \times 11^1$

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QuestionFind the prime factors of the following numbers and express them in the index form : 4.95 2.2303. 105 8.110

Studdy Solution

STEP 1

1. Prime factorization involves breaking down a number into its prime factors.2. The prime factors of a number are the prime numbers that divide the number exactly, without leaving a remainder.3. Index form (or exponential form) involves expressing repeated prime factors using exponents.

STEP 2

1. Prime factorize the number 95.2. Prime factorize the number 230.3. Prime factorize the number 105.4. Prime factorize the number 110.

STEP 3

Determine if 95 is divisible by the smallest prime number, which is 2. Since 95 is odd, it is not divisible by 2.

STEP 4

Check the next smallest prime number: 3. The sum of the digits of 95 is 9+5=149+5=149+5=14, which is not divisible by 3. Hence, 95 is not divisible by 3.

STEP 5

Check the next smallest prime number: 5. Since 95 ends in 5, it is divisible by 5. 95÷5=19 95 \div 5 = 19 95÷5=19

STEP 6

Determine if 19 is a prime number. Since 19 is not divisible by 2, 3, or 5 (the prime numbers less than its square root), 19 is a prime number.

STEP 7

Express the prime factors of 95 in index form: 95=51×191 95 = 5^1 \times 19^1 95=51×191

STEP 8

Determine if 230 is divisible by the smallest prime number, which is 2. Since 230 is even, it is divisible by 2. 230÷2=115 230 \div 2 = 115 230÷2=115

STEP 9

Check if 115 is divisible by the next smallest prime number: 3. The sum of the digits of 115 is 1+1+5=71+1+5=71+1+5=7, which is not divisible by 3. Hence, 115 is not divisible by 3.

STEP 10

Check the next smallest prime number: 5. Since 115 ends in 5, it is divisible by 5. 115÷5=23 115 \div 5 = 23 115÷5=23

STEP 11

Determine if 23 is a prime number. Since 23 is not divisible by 2, 3, or 5 (the prime numbers less than its square root), 23 is a prime number.

STEP 12

Express the prime factors of 230 in index form: 230=21×51×231 230 = 2^1 \times 5^1 \times 23^1 230=21×51×231

STEP 13

Determine if 105 is divisible by the smallest prime number, which is 2. Since 105 is odd, it is not divisible by 2.

STEP 14

Check the next smallest prime number: 3. The sum of the digits of 105 is 1+0+5=61+0+5=61+0+5=6, which is divisible by 3. Hence, 105 is divisible by 3. 105÷3=35 105 \div 3 = 35 105÷3=35

STEP 15

Check if 35 is divisible by the next smallest prime number: 5. Since 35 ends in 5, it is divisible by 5. 35÷5=7 35 \div 5 = 7 35÷5=7

STEP 16

Determine if 7 is a prime number. Since 7 is not divisible by 2 or 3 (the prime numbers less than its square root), 7 is a prime number.

STEP 17

Express the prime factors of 105 in index form: 105=31×51×71 105 = 3^1 \times 5^1 \times 7^1 105=31×51×71

STEP 18

Determine if 110 is divisible by the smallest prime number, which is 2. Since 110 is even, it is divisible by 2. 110÷2=55 110 \div 2 = 55 110÷2=55

STEP 19

Check if 55 is divisible by the next smallest prime number: 3. The sum of the digits of 55 is 5+5=105+5=105+5=10, which is not divisible by 3. Hence, 55 is not divisible by 3.

STEP 20

Check the next smallest prime number: 5. Since 55 ends in 5, it is divisible by 5. 55÷5=11 55 \div 5 = 11 55÷5=11

STEP 21

Determine if 11 is a prime number. Since 11 is not divisible by 2, 3, or 5 (the prime numbers less than its square root), 11 is a prime number.

STEP 22

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QuestionFind the prime factors of the following numbers and express them in the index form : 4.95 2.230
3. 105 8.110

1. Prime factorization involves breaking down a number into its prime factors.
2. The prime factors of a number are the prime numbers that divide the number exactly, without leaving a remainder.
3. Index form (or exponential form) involves expressing repeated prime factors using exponents.

1. Prime factorize the number 95.
2. Prime factorize the number 230.
3. Prime factorize the number 105.
4. Prime factorize the number 110.

Check the next smallest prime number: 3. The sum of the digits of 95 is $9+5=14$ , which is not divisible by 3. Hence, 95 is not divisible by 3.

Check the next smallest prime number: 5. Since 95 ends in 5, it is divisible by 5. $95 \div 5 = 19$

Express the prime factors of 95 in index form: $95 = 5^1 \times 19^1$

Determine if 230 is divisible by the smallest prime number, which is 2. Since 230 is even, it is divisible by 2. $230 \div 2 = 115$

Check if 115 is divisible by the next smallest prime number: 3. The sum of the digits of 115 is $1+1+5=7$ , which is not divisible by 3. Hence, 115 is not divisible by 3.

Check the next smallest prime number: 5. Since 115 ends in 5, it is divisible by 5. $115 \div 5 = 23$

Express the prime factors of 230 in index form: $230 = 2^1 \times 5^1 \times 23^1$

Check the next smallest prime number: 3. The sum of the digits of 105 is $1+0+5=6$ , which is divisible by 3. Hence, 105 is divisible by 3. $105 \div 3 = 35$

Check if 35 is divisible by the next smallest prime number: 5. Since 35 ends in 5, it is divisible by 5. $35 \div 5 = 7$

Express the prime factors of 105 in index form: $105 = 3^1 \times 5^1 \times 7^1$

Determine if 110 is divisible by the smallest prime number, which is 2. Since 110 is even, it is divisible by 2. $110 \div 2 = 55$

Check if 55 is divisible by the next smallest prime number: 3. The sum of the digits of 55 is $5+5=10$ , which is not divisible by 3. Hence, 55 is not divisible by 3.

Check the next smallest prime number: 5. Since 55 ends in 5, it is divisible by 5. $55 \div 5 = 11$