Number System Question 6 - CAT 2025

Step-by-Step Explanation

• Convert the expression to base prime components:

Rewrite both sides of the equation by breaking the bases down into powers of 2, 3, and 5:

LHS = ( 2² × 3 )¹²ˣ × ( 2² )²⁴ˣ⁺¹² × 5²ʸ = 2²⁴ˣ × 3¹²ˣ × 2⁴⁸ˣ⁺²⁴ × 5²ʸ

RHS = ( 2² )⁶ᶻ × ( 2² × 5 )¹²ˣ × ( 3⁵ )³ˣ⁻⁶ = 2¹²ᶻ × 2²⁴ˣ × 5¹²ˣ × 3¹⁵ˣ⁻³⁰

• Equate corresponding prime exponents:

Since the equation must balance perfectly, match the exponential variables for each prime base independently:

Powers of 3: 12x = 15x - 30 ⇒ 3x = 30 ⇒ x = 10

Powers of 5: 2y = 12x ⇒ y = 6x ⇒ y = 6(10) = 60

Powers of 2: 24x + 48x + 24 = 12z + 24x ⇒ 48x + 24 = 12z ⇒ 4x + 2 = z

z = 4(10) + 2 = 42

• Sum the final values:

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

Correct Answer: 112