Question

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

Answer 1

The prime factorizations of 1134 and 168 are:
\(168 = 2^3 × 3 × 7\)
1134 = \(2 × 3^4 × 7.\)

The smallest positive integer n for which 168 is a factor of 1134ⁿ is 3.
\(1134^3 = 2^3 × 3^{12} × 7^3 = 1134^n \)
The smallest positive integer m for which 1134³ is a factor of 168^m is 12.
Therefore, \(m + n = 12 + 3 = 15\).
The correct answer is 15.