Question

Let $n$ be the least positive integer such that $168$ is a factor of $1134^n$ . If $m$ is the least positive integer such that $1134^n$ is a factor of $168^m$ , then $m+ n$ equals

• A. 15
• B. 12
• C. 24
• D. 9

Answer 1

Answer: (A)

15

Answer 2

The following are the prime factorizations of 1134 and 168:
$168 = 2^3 × 3 × 7$
$1134 = 2 × 3^4 × 7$

Clearly, 3 is the least positive integral number of n that allows 168 to be a factor of $1134^n$.
$1134^3 = 2^3 × 3^{12} × 7^3 = 1134^n$
It is evident that 12 is the least positive integral value of m that allows $1134^3$ to be a factor of $168^m.$
It follows that $m + n = 12 + 3 = 15$
The correct option is (A): 15