A positive integer n is a multiple of 7, and the greatest common factor of n and 120 is 15… — ACT

A positive integer $n$ is a multiple of $7$ , and the greatest common factor of $n$ and $120$ is $15$ . Which of the following is a possible value for $n$ ?

A
75
105Answer
C
120
D
210
E
420

Answer

105

The correct answer is 105. Since the greatest common factor of

n

and 120 is 15,

n

must be a multiple of 15. We are also given that

n

is a multiple of 7. Since 7 and 15 share no common factors other than 1,

n

must be a multiple of

7 \times 15 = 105

. Let

n = 105k

for some positive integer

k

. We must ensure that the greatest common factor of

105k

and 120 is exactly 15. Since

\text{GCF}(105k, 120) = 15 \times \text{GCF}(7k, 8)

, we require

\text{GCF}(7k, 8) = 1

, which means

k

must be an odd integer. Choosing the smallest positive odd integer

k = 1

yields

n = 105

, which is a possible value.

Step-by-Step Solution

Analyze the prime factorization of 120 and the greatest common factor requirement.

Since the greatest common factor (

\text{GCF}

) of

n

and 120 is 15,

n

must be divisible by 15. The prime factorization of 120 is

2^3 \times 3 \times 5

, and

15 = 3 \times 5

. Therefore,

n

must contain the prime factors 3 and 5, but cannot contain any powers of 2.

To establish which prime factors

n

must and must not have to satisfy the GCF condition.

Combine the GCF divisibility condition with the multiple of 7 constraint.

Since

n

is a multiple of 7 and also a multiple of 15, and because 7 and 15 are coprime (they share no common prime factors),

n

must be a multiple of

7 \times 15 = 105

. Let

n = 105k

for some positive integer

k

To find the general form of

n

by finding the least common multiple of the required factor components.

Test the constraints on

k

to identify valid values for

n

We need

\text{GCF}(105k, 120) = 15

. This can be written as

\text{GCF}(15 \times 7k, 15 \times 8) = 15 \times \text{GCF}(7k, 8) = 15

, which requires

\text{GCF}(7k, 8) = 1

. Since 7 and 8 are coprime,

k

must not share any common factors with 8 (i.e.,

k

must be odd). The smallest positive odd integer is

k = 1

, which gives

n = 105

To determine the valid values of

n

and check them against the options.

Key Concept

Using prime factorizations to determine greatest common factors under multiple divisibility constraints.

Estimated Time:1m 0s

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