CAT 2025

Slot 3

QA

Question & SolutionQues & Sol

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

1. Concept Used

• Topic: Number Theory — Prime Factorisation & Comparing Exponents
• Formula: $$ a^m = a^n \implies m = n \quad (\text{when bases are equal and prime}) $$

2. Calculation

We start by prime-factorising every term on both sides of the equation:

$12^{12x} \times 4^{24x+12} \times 5^{2y} = 8^{4z} \times 20^{12x} \times 243^{3x-6}$

Breaking each base into primes:

• $12 = 2^2 \times 3$, so $12^{12x} = 2^{24x} \times 3^{12x}$
• $4 = 2^2$, so $4^{24x+12} = 2^{48x+24}$
• $8 = 2^3$, so $8^{4z} = 2^{12z}$
• $20 = 2^2 \times 5$, so $20^{12x} = 2^{24x} \times 5^{12x}$
• $243 = 3^5$, so $243^{3x-6} = 3^{15x-30}$

Substituting back:

$\text{LHS} = 2^{24x} \times 3^{12x} \times 2^{48x+24} \times 5^{2y} = 2^{72x+24} \times 3^{12x} \times 5^{2y}$

$\text{RHS} = 2^{12z} \times 2^{24x} \times 5^{12x} \times 3^{15x-30} = 2^{12z+24x} \times 3^{15x-30} \times 5^{12x}$

Now equating the powers of each prime independently:

Powers of 3: $12x = 15x - 30 \implies 3x = 30 \implies x = 10$

Powers of 5: $2y = 12x \implies 2y = 12 \times 10 = 120 \implies y = 60$

Powers of 2: $72x + 24 = 12z + 24x \implies 48x + 24 = 12z \implies z = 4x + 2 = 4(10) + 2 = 42$

Therefore: $x + y + z = 10 + 60 + 42 = 112$

3. Solution

Answer = 112 ✅

The final calculated value is 112.