Question

Find the sum of digits of the number \(625^{65} \times 128^{36}\).

• A. 20
• B. 25
• C. 30
• D. 35
• E. 40

Hint:

For problems involving the sum of digits of large numbers raised to powers, always try to simplify the expression by finding common bases or forming powers of 10. This avoids calculating the gigantic number itself.

Answer 1

The Correct Option is B

Step 1: Prime-factorise: \( 625^{65}\times128^{36} = 5^{260}\times2^{252} \). Matching 252 fives with the 252 twos leaves a surplus of \( 260-252=8 \) unpaired fives, so the number is \( 5^{8} \) followed by 252 zeros.
Step 2: Build \( 5^{8} \) by repeated multiplication instead of squaring: \( 5,\ 25,\ 125,\ 625,\ 3125,\ 15625,\ 78125,\ 390625 \) (each term is the previous one times 5, eight terms in all).
Step 3: So \( 5^{8}=390625 \), and the digit sum of 390625 (the zeros contribute nothing) is:
\[ 3+9+0+6+2+5 = \boxed{25}. \]

Final answer: 25 (option B).