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 168^m. 
It follows that \(m + n = 12 + 3 = 15\) 
The correct answer is 15.