Number System Question 3 - CAT 2025

Step-by-Step Explanation

• Perform prime factorization to reveal powers of 10:

Convert the composite bases into their respective prime component structures:

625 = 5⁴ ⇒ (625)⁶⁵ = ( 5⁴ )⁶⁵ = 5²⁶⁰

128 = 2⁷ ⇒ (128)³⁶ = ( 2⁷ )³⁶ = 2²⁵²

• Combine the base components:

(625)⁶⁵ × (128)³⁶ = 5²⁶⁰ × 2²⁵² = 5⁸ × (5 × 2)²⁵² = 5⁸ × 10²⁵²

Evaluate the leading non-zero coefficient:

5⁸ = 390625

The full expansion forms the number 390625 followed by 252 terminal zeros.

• Sum the individual digits:

Since zeros do not contribute to the final sum:

Sum of digits = 3 + 9 + 0 + 6 + 2 + 5 = 25

Correct Answer: 25

💡 FundaMakers Tip:

Large products containing combinations of 2 and 5 are testing your ability to isolate base-10 indices. Pair the matching exponents together as (2 × 5)ᵏ to drop the trailing values into meaningless zeros, leaving a simple leading integer to evaluate.