CAT 2025

Slot 2

QA

Question & Solution

1. Concept Used

• Topic: Number Systems — Prime Factorization and Sum of Digits
• Formula: $$2^a \times 5^a = 10^a \quad \text{(pairing powers of 2 and 5 to form trailing zeros)}$$

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2. Calculation

First, express both bases as powers of their prime factors.

$$625 = 5^4 \quad \text{and} \quad 128 = 2^7$$

So the expression becomes:

$$(625)^{65} \times (128)^{36} = (5^4)^{65} \times (2^7)^{36} = 5^{260} \times 2^{252}$$

Now, pair equal powers of 2 and 5 to form powers of 10. Since the power of 2 is the smaller one (252), we can extract $$10^{252}$$:

$$5^{260} \times 2^{252} = 5^{252} \times 2^{252} \times 5^{260-252} = 10^{252} \times 5^8$$

This means the number is $$5^8$$ followed by 252 trailing zeros.

Now compute $$5^8$$:

$$5^1 = 5$$ $$5^2 = 25$$ $$5^4 = 625$$ $$5^8 = 625^2 = 390625$$

So the full number is: $$390625\underbrace{00\cdots0}_{252 \text{ zeros}}$$

The sum of digits is contributed only by $$390625$$ (zeros contribute nothing):

$$3 + 9 + 0 + 6 + 2 + 5 = 25$$

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3. Solution

Answer = 25 ✅

The final calculated sum of digits is 25.