Question 4.

Prime factorising 1134, we get  $1134 = 2 \times 3^4 \times 7$  and  $168 = 2^3 \times 3 \times 7$

$1134^n$ is a factor of 168  $\Rightarrow$ the factor of 2 should be at least 3, for 168 to be a factor  $\Rightarrow n = 3$

Now,  $1134^n = 1134^3 = 2^3 \times 3^{12} \times 7^3$  is a factor of  $168^m = \left(2^3 \times 3 \times 7\right)^m$  $\Rightarrow m = 12$ as power of 3 should be at least 12.

$\Rightarrow$ So, $m + n = 15$