CAT 2025 Slot 2QANumber Systems • MiscellaneousModerate
The sum of digits of the number $(625)^{65} × (128)^{36} , is
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Official Correct Answer: 25. ### Core Concept
The problem involves finding the sum of the digits of a large number formed by multiplying two exponentials. By expressing the numbers in terms of their prime factors, we can simplify the expression and compute the digit sum efficiently.
### Step-by-Step Solution
1. **Factorize the Numbers**:
- \(625 = 5^4\)
- \(128 = 2^7\)
2. **Express the Exponentials**:
- \((625)^{65} = (5^4)^{65} = 5^{260}\)
- \((128)^{36} = (2^7)^{36} = 2^{252}\)
3. **Combine the Factors**:
- Multiply the two results: \(5^{260} \times 2^{252}\)
- This can be rewritten as \(10^{252} \times 5^8\) because \(5^{252} \times 2^{252} = 10^{252}\) and \(5^{260} = 5^{252} \times 5^8\)
4. **Compute \(5^8\)**:
- \(5^8 = 390625\)
5. **Form the Number**:
- The number is \(390625 \times 10^{252}\), which is 390625 followed by 252 zeros.
6. **Sum the Digits**:
- The non-zero digits are 3, 9, 0, 6, 2, 5.
- Sum: \(3 + 9 + 0 + 6 + 2 + 5 = 25\)
### 30-Second Shortcut
- Break down the numbers into prime factors to simplify the exponentials.
- Recognize that multiplying by \(10^{252}\) appends 252 zeros, which don't affect the digit sum.
- Sum only the non-zero digits of the coefficient.
### Final Answer
The sum of the digits is 25.
**Correct Answer: 25**