Question

(1134)n is divisible by 168. (168)m is divisible by (1134)n. Find the minimum value of n + m

Answer 1

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

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