Solved on Feb 08, 2024

Determine if each number is prime or composite, and explain your reasoning. $27, 19, 31, 38, 45, 53, 87, 93$

STEP 1

Assumptions
1. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
2. A composite number is a natural number greater than 1 that has more than two positive divisors.
3. We will use divisibility rules and prime factorization to determine if a number is prime or composite.

STEP 2

To determine if the number 27 is prime or composite, we can use the fact that any number greater than 1 that is not prime is composite.
$27 = 3 \times 3 \times 3$

STEP 3

Since 27 can be divided exactly by 3, it is not a prime number.
$27 \text{ is composite}$

STEP 4

To determine if the number 19 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{19} \approx 4.36$

STEP 5

The prime numbers less than 4.36 are 2 and 3. Since 19 is not divisible by 2 or 3, we conclude that 19 is a prime number.
$19 \text{ is prime}$

STEP 6

To determine if the number 31 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{31} \approx 5.57$

STEP 7

The prime numbers less than 5.57 are 2, 3, and 5. Since 31 is not divisible by 2, 3, or 5, we conclude that 31 is a prime number.
$31 \text{ is prime}$

STEP 8

To determine if the number 38 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{38} \approx 6.16$

STEP 9

The prime numbers less than 6.16 are 2, 3, and 5. Since 38 is divisible by 2 (38 = 2 x 19), we conclude that 38 is a composite number.
$38 \text{ is composite}$

STEP 10

To determine if the number 45 is prime or composite, we can use the fact that it ends in a 5, which means it is divisible by 5.
$45 = 5 \times 9$

STEP 11

Since 45 can be divided exactly by 5, it is not a prime number.
$45 \text{ is composite}$

STEP 12

To determine if the number 53 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{53} \approx 7.28$

STEP 13

The prime numbers less than 7.28 are 2, 3, 5, and 7. Since 53 is not divisible by 2, 3, 5, or 7, we conclude that 53 is a prime number.
$53 \text{ is prime}$

STEP 14

To determine if the number 87 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{87} \approx 9.33$

STEP 15

The prime numbers less than 9.33 are 2, 3, 5, and 7. Since 87 is divisible by 3 (87 = 3 x 29), we conclude that 87 is a composite number.
$87 \text{ is composite}$

STEP 16

To determine if the number 93 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{93} \approx 9.64$

STEP 17

The prime numbers less than 9.64 are 2, 3, 5, and 7. Since 93 is divisible by 3 (93 = 3 x 31), we conclude that 93 is a composite number.
$93 \text{ is composite}$
The numbers 19, 31, and 53 are prime, while the numbers 27, 38, 45, 87, and 93 are composite.

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Determine if each number is prime or composite, and explain your reasoning. $27, 19, 31, 38, 45, 53, 87, 93$

STEP 1

STEP 2

To determine if the number 27 is prime or composite, we can use the fact that any number greater than 1 that is not prime is composite.
$27 = 3 \times 3 \times 3$

STEP 3

Since 27 can be divided exactly by 3, it is not a prime number.
$27 \text{ is composite}$

STEP 4

To determine if the number 19 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{19} \approx 4.36$

STEP 5

The prime numbers less than 4.36 are 2 and 3. Since 19 is not divisible by 2 or 3, we conclude that 19 is a prime number.
$19 \text{ is prime}$

STEP 6

To determine if the number 31 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{31} \approx 5.57$

STEP 7

The prime numbers less than 5.57 are 2, 3, and 5. Since 31 is not divisible by 2, 3, or 5, we conclude that 31 is a prime number.
$31 \text{ is prime}$

STEP 8

To determine if the number 38 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{38} \approx 6.16$

STEP 9

The prime numbers less than 6.16 are 2, 3, and 5. Since 38 is divisible by 2 (38 = 2 x 19), we conclude that 38 is a composite number.
$38 \text{ is composite}$

STEP 10

To determine if the number 45 is prime or composite, we can use the fact that it ends in a 5, which means it is divisible by 5.
$45 = 5 \times 9$

STEP 11

Since 45 can be divided exactly by 5, it is not a prime number.
$45 \text{ is composite}$

STEP 12

To determine if the number 53 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{53} \approx 7.28$

STEP 13

The prime numbers less than 7.28 are 2, 3, 5, and 7. Since 53 is not divisible by 2, 3, 5, or 7, we conclude that 53 is a prime number.
$53 \text{ is prime}$

STEP 14

To determine if the number 87 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{87} \approx 9.33$

STEP 15

The prime numbers less than 9.33 are 2, 3, 5, and 7. Since 87 is divisible by 3 (87 = 3 x 29), we conclude that 87 is a composite number.
$87 \text{ is composite}$

STEP 16

To determine if the number 93 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{93} \approx 9.64$

STEP 17

The prime numbers less than 9.64 are 2, 3, 5, and 7. Since 93 is divisible by 3 (93 = 3 x 31), we conclude that 93 is a composite number.
$93 \text{ is composite}$
The numbers 19, 31, and 53 are prime, while the numbers 27, 38, 45, 87, and 93 are composite.

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Determine if each number is prime or composite, and explain your reasoning. 27,19,31,38,45,53,87,9327, 19, 31, 38, 45, 53, 87, 9327,19,31,38,45,53,87,93

STEP 1

STEP 2

To determine if the number 27 is prime or composite, we can use the fact that any number greater than 1 that is not prime is composite.27=3×3×327 = 3 \times 3 \times 327=3×3×3

STEP 3

Since 27 can be divided exactly by 3, it is not a prime number.27 is composite27 \text{ is composite}27 is composite

STEP 4

To determine if the number 19 is prime or composite, we check for divisibility by prime numbers less than its square root.19≈4.36\sqrt{19} \approx 4.3619​≈4.36

STEP 5

The prime numbers less than 4.36 are 2 and 3. Since 19 is not divisible by 2 or 3, we conclude that 19 is a prime number.19 is prime19 \text{ is prime}19 is prime

STEP 6

To determine if the number 31 is prime or composite, we check for divisibility by prime numbers less than its square root.31≈5.57\sqrt{31} \approx 5.5731​≈5.57

STEP 7

The prime numbers less than 5.57 are 2, 3, and 5. Since 31 is not divisible by 2, 3, or 5, we conclude that 31 is a prime number.31 is prime31 \text{ is prime}31 is prime

STEP 8

To determine if the number 38 is prime or composite, we check for divisibility by prime numbers less than its square root.38≈6.16\sqrt{38} \approx 6.1638​≈6.16

STEP 9

The prime numbers less than 6.16 are 2, 3, and 5. Since 38 is divisible by 2 (38 = 2 x 19), we conclude that 38 is a composite number.38 is composite38 \text{ is composite}38 is composite

STEP 10

To determine if the number 45 is prime or composite, we can use the fact that it ends in a 5, which means it is divisible by 5.45=5×945 = 5 \times 945=5×9

STEP 11

Since 45 can be divided exactly by 5, it is not a prime number.45 is composite45 \text{ is composite}45 is composite

STEP 12

To determine if the number 53 is prime or composite, we check for divisibility by prime numbers less than its square root.53≈7.28\sqrt{53} \approx 7.2853​≈7.28

STEP 13

The prime numbers less than 7.28 are 2, 3, 5, and 7. Since 53 is not divisible by 2, 3, 5, or 7, we conclude that 53 is a prime number.53 is prime53 \text{ is prime}53 is prime

STEP 14

To determine if the number 87 is prime or composite, we check for divisibility by prime numbers less than its square root.87≈9.33\sqrt{87} \approx 9.3387​≈9.33

STEP 15

The prime numbers less than 9.33 are 2, 3, 5, and 7. Since 87 is divisible by 3 (87 = 3 x 29), we conclude that 87 is a composite number.87 is composite87 \text{ is composite}87 is composite

STEP 16

To determine if the number 93 is prime or composite, we check for divisibility by prime numbers less than its square root.93≈9.64\sqrt{93} \approx 9.6493​≈9.64

STEP 17

The prime numbers less than 9.64 are 2, 3, 5, and 7. Since 93 is divisible by 3 (93 = 3 x 31), we conclude that 93 is a composite number.93 is composite93 \text{ is composite}93 is compositeThe numbers 19, 31, and 53 are prime, while the numbers 27, 38, 45, 87, and 93 are composite.

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Determine if each number is prime or composite, and explain your reasoning. $27, 19, 31, 38, 45, 53, 87, 93$

To determine if the number 27 is prime or composite, we can use the fact that any number greater than 1 that is not prime is composite.
$27 = 3 \times 3 \times 3$

Since 27 can be divided exactly by 3, it is not a prime number.
$27 \text{ is composite}$

To determine if the number 19 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{19} \approx 4.36$

The prime numbers less than 4.36 are 2 and 3. Since 19 is not divisible by 2 or 3, we conclude that 19 is a prime number.
$19 \text{ is prime}$

To determine if the number 31 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{31} \approx 5.57$

The prime numbers less than 5.57 are 2, 3, and 5. Since 31 is not divisible by 2, 3, or 5, we conclude that 31 is a prime number.
$31 \text{ is prime}$

To determine if the number 38 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{38} \approx 6.16$

The prime numbers less than 6.16 are 2, 3, and 5. Since 38 is divisible by 2 (38 = 2 x 19), we conclude that 38 is a composite number.
$38 \text{ is composite}$

To determine if the number 45 is prime or composite, we can use the fact that it ends in a 5, which means it is divisible by 5.
$45 = 5 \times 9$

Since 45 can be divided exactly by 5, it is not a prime number.
$45 \text{ is composite}$

To determine if the number 53 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{53} \approx 7.28$

The prime numbers less than 7.28 are 2, 3, 5, and 7. Since 53 is not divisible by 2, 3, 5, or 7, we conclude that 53 is a prime number.
$53 \text{ is prime}$

To determine if the number 87 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{87} \approx 9.33$

The prime numbers less than 9.33 are 2, 3, 5, and 7. Since 87 is divisible by 3 (87 = 3 x 29), we conclude that 87 is a composite number.
$87 \text{ is composite}$

To determine if the number 93 is prime or composite, we check for divisibility by prime numbers less than its square root.
$\sqrt{93} \approx 9.64$

The prime numbers less than 9.64 are 2, 3, 5, and 7. Since 93 is divisible by 3 (93 = 3 x 31), we conclude that 93 is a composite number.
$93 \text{ is composite}$
The numbers 19, 31, and 53 are prime, while the numbers 27, 38, 45, 87, and 93 are composite.