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CAT 2025Slot 2QAQues & Sol

Number SystemsMedium

Question

The sum of digits of the number $(625)^{65} \times (128)^{36}$ is

Solution

1. Concept Used

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

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):

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

3. Solution

Answer = 25 ✅

The final calculated sum of digits is 25.