CAT 2023 Slot 1 QA Question 3: Let n be the least positive integer such that 168 is a factor of $1134^n$. If m is the least positive integer such that $1134^n$ is a factor of $168^m$, then $m+n$ equals

Question

Let n be the least positive integer such that 168 is a factor of $1134^n$. If m is the least positive integer such that $1134^n$ is a factor of $168^m$, then $m+n$ equals

Accepted Answer

Prime factorise both numbers.

$1134 = 2 	imes 3^4 	imes 7$
$168 = 2^3 	imes 3 	imes 7$

For 168 to divide $1134^n$, the exponent of 2 in $1134^n$ must be at least 3.
$n \geq 3$
Therefore, the least possible value is $n=3$.

Now, $1134^3 = 2^3 	imes 3^{12} 	imes 7^3$ must divide $168^m = 2^{3m

Let n be the least positive integer such that 168 is a factor of 1134n1134^n1134n. If m is the least positive integer such that 1134n1134^n1134n is a factor of 168m168^m168m, then m+nm+nm+n equals

Let n be the least positive integer such that 168 is a factor of $1134^n$ . If m is the least positive integer such that $1134^n$ is a factor of $168^m$ , then $m+n$ equals