AAPL wrote:Manhattan Prep

\(m = 4n + 9\), where \(n\) is a positive integer. What is the greatest common factor of \(m\) and \(n\)?

1. \(m=9s\), where \(s\) is a positive integer.
2. \(n=4t\), where \(t\)is a positive integer.

OA A

We are asked to determine the greatest common factor (GCF) of m and n, given that m = 4n + 9.

Statement One Alone:

m = 9s, where s is a positive integer.

Since m = 4n + 9, we have:

9s = 4n + 9

9s - 9 = 4n

9(s - 1) = 4n

Since 4 is not divisible by 9, n must be divisible by 9. That is, n = 9k for some positive integer k. In that case, m = 4(9k) + 9 = 36k + 9 = 9(4k + 1). We see that m is also divisible by 9. Since k and 4k + 1 are relatively prime, then the GCF of m and n is 9. Statement one alone is sufficient.

Statement Two Alone:

n = 4t, where t is a positive integer.

Since m = 4n + 9, we have:

m = 4(4t) + 9

m = 16t + 9

This is not sufficient to determine the GCF of m and n. For example, if t = 1, n = 4(1) = 4 and m = 16(1) + 9 = 25 and GCF(4, 25) = 1. However, if t = 9, then n = 4(9) = 36 and m = 16(9) + 9 = 153 and GCF(36, 153) = 9. Statement two alone is not sufficient.

Answer: A