CAT 2025

Slot 3

QA

Question & Solution

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

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3. Solution

Answer = 112 ✅

The final calculated value is 112.