Question

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

Hint:

Whenever you see an equation involving different bases with exponents, convert all bases to prime factors. Then compare the exponents of each prime on both sides — it turns the problem into a simple system of linear equations.

Answer 1

Correct Answer: 112

Step 1: Express both sides in prime bases 2, 3, 5. LHS \( = 2^{72x+24}\cdot3^{12x}\cdot5^{2y} \); RHS \( = 2^{24x+12z}\cdot3^{15x-30}\cdot5^{12x} \).
Step 2: Equate powers of 3: \( 12x=15x-30 \Rightarrow x=10 \).
Step 3: Equate powers of 5: \( 2y=12x \Rightarrow y=60 \). Equate powers of 2: \( 72x+24=24x+12z \Rightarrow z=42 \).
\[ \boxed{x+y+z = 112} \]