Number Theory and Fractions Unit Test

Question

1 of 151 of 15 Items
Question
Determine the number greater than 70 but less than 85 that is divisible by both 2 and 9.(1 point)
Responses

78
78

81
81

72
72

84

Answer 1

To find a number greater than 70 but less than 85 that is divisible by both 2 and 9, we first identify the multiples of 2 and 9 within that range.

• The multiples of 2 between 70 and 85 are: 72, 74, 76, 78, 80, 82, 84.
• The multiples of 9 between 70 and 85 are: 72, 81, 90 (but 90 is out of the range).

Next, we find the common multiples from both lists:

• From the multiples of 2, we have: 72, 74, 76, 78, 80, 82, 84.
• From the multiples of 9, we have: 72, 81.

The only number that appears in both lists and is between 70 and 85 is 72.

Thus, the correct answer is 72.

Answer 2

Jesse lists four factor pairs of 72 as 2 × 36
, 4 × 18
, 6 × 12
, and 8 × 9
. Which factor pair has a prime number?(1 point)
Responses

6 × 12
6 times 12

4 × 18
4 times 18

8 × 9
8 times 9

2 × 36

Answer 3

To identify which factor pair of 72 contains a prime number, we can analyze each factor pair:

1. 2 × 36: The factors are 2 and 36. The number 2 is a prime number.
2. 4 × 18: The factors are 4 and 18. Neither of these numbers is prime (4 is 2 × 2 and 18 is 2 × 9).
3. 6 × 12: The factors are 6 and 12. Neither of these is prime (6 is 2 × 3 and 12 is 2 × 6).
4. 8 × 9: The factors are 8 and 9. Neither of these is prime (8 is 2 × 4 and 9 is 3 × 3).

From the analysis, the factor pair 2 × 36 contains the prime number 2.

Therefore, the correct answer is 2 × 36.

Answer 4

The composite number 50 is between what two prime numbers?(1 point)
The composite number 50 lies between
and
, both prime numbers.

Answer 5

The composite number 50 lies between the prime numbers 47 and 53.

So, the answer is: The composite number 50 lies between 47 and 53, both prime numbers.

Answer 6

Express the prime factorization of 1,200 as a product of prime numbers with exponents.(2 points)
4
⋅

⋅

2
= 1,200

Answer 7

To express the prime factorization of 1,200, we can start by dividing it by the smallest prime number, 2, and continue factoring until we are left with only prime numbers.

1. Divide by 2: \[ 1200 \div 2 = 600 \ 600 \div 2 = 300 \ 300 \div 2 = 150 \ 150 \div 2 = 75 \]

   Now, we can no longer divide by 2 since 75 is not even.

2. Next, we check the next prime number, which is 3: \[ 75 \div 3 = 25 \]

3. Now, we factor 25, which is \(5 \times 5\): \[ 25 \div 5 = 5 \ 5 \div 5 = 1 \]

Now, putting this all together, we have: \[ 1200 = 2^4 \times 3^1 \times 5^2 \]

Thus, the prime factorization of 1,200 as a product of prime numbers with exponents is: \[ 2^4 \cdot 3^1 \cdot 5^2 = 1200 \]