Divisibility and Prime Factorization

Prime factorizations: $168 = 2^3 \times 3 \times 7$ $1134 = 2 \times 3^4 \times 7$

For $168$ to divide $1134^n = 2^n \times 3^{4n} \times 7^n$:

• Power of $2$: $n \ge 3$
• Power of $3$: $4n \ge 1 \implies n \ge 1$
• Power of $7$: $n \ge 1$ Thus, the least positive integer $n = 3$.

So $1134^n = 1134^3 = 2^3 \times 3^{12} \times 7^3$.

For $1134^3$ to divide $168^m = 2^{3m} \times 3^m \times 7^m$:

• Power of $2$: $3m \ge 3 \implies m \ge 1$
• Power of $3$: $m \ge 12$
• Power of $7$: $m \ge 3$ Thus, the least positive integer $m = 12$.

Therefore, $m + n = 12 + 3 = 15$.