CAT 2025 Slot 3 QA Question 5: If $12^{12x} imes 4^{24x + 12} imes 5^{2y} = 8^{4z} imes 20^{12x} imes 243^{3x - 6}$, where $x$, $y$ and $z$ are natural numbers, then $x + y + z$ equals

Question

If $12^{12x} 	imes 4^{24x + 12} 	imes 5^{2y} = 8^{4z} 	imes 20^{12x} 	imes 243^{3x - 6}$, where $x$, $y$ and $z$ are natural numbers, then $x + y + z$ equals

Accepted Answer

Prime-factorise both sides and equate the powers of the same primes.
$12 = 2^{2} 	imes 3, 4 = 2^{2}, 8 = 2^{3}, 20 = 2^{2} 	imes 5, 243 = 3^{5}$
$	ext{LHS} = 2^{72x+24} 	imes 3^{12x} 	imes 5^{2y}$
$	ext{RHS} = 2^{12z+24x} 	imes 3^{15x-30} 	imes 5^{12x}$
Equating powers of $3$ gives
$12x = 15

If 1212x×424x+12×52y=84z×2012x×2433x−612^{12x} \times 4^{24x + 12} \times 5^{2y} = 8^{4z} \times 20^{12x} \times 243^{3x - 6}1212x×424x+12×52y=84z×2012x×2433x−6, where xxx, yyy and zzz are natural numbers, then x+y+zx + y + zx+y+z equals

If $12^{12x} \times 4^{24x + 12} \times 5^{2y} = 8^{4z} \times 20^{12x} \times 243^{3x - 6}$ , where $x$ , $y$ and $z$ are natural numbers, then $x + y + z$ equals