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: Understanding the Question:
The problem asks for the sum of the digits of the result of the calculation \(625^{65} \times 128^{36}\).
Step 2: Key Formula or Approach:
The key to solving this problem is to express the bases of the exponents, 625 and 128, in terms of prime factors. Specifically, we aim to create powers of 10, since multiplying by \(10^n\) simply adds \(n\) zeros to a number without changing the sum of its digits. We know that \(10 = 2 \times 5\).
- \(625 = 5 \times 5 \times 5 \times 5 = 5^4\)
- \(128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7\)
Step 3: Detailed Explanation:
First, substitute the prime factor forms into the original expression:
\[ (5^4)^{65} \times (2^7)^{36} \] Using the power of a power rule \((a^m)^n = a^{mn}\), we simplify the exponents:
\[ 5^{4 \times 65} \times 2^{7 \times 36} \] \[ 5^{260} \times 2^{252} \] To form powers of 10, we need to pair up the 2s and 5s. We can rewrite \(5^{260}\) as \(5^{8} \times 5^{252}\):
\[ (5^8 \times 5^{252}) \times 2^{252} \] Group the terms with the same exponent:
\[ 5^8 \times (5^{252} \times 2^{252}) \] Using the rule \(a^n \times b^n = (ab)^n\):
\[ 5^8 \times (5 \times 2)^{252} \] \[ 5^8 \times 10^{252} \] Now, we need to calculate the value of \(5^8\):
- \(5^2 = 25\)
- \(5^4 = (5^2)^2 = 25^2 = 625\)
- \(5^8 = (5^4)^2 = 625^2 = 390625\)
So, the number is \(390625 \times 10^{252}\). This represents the number 390625 followed by 252 zeros.
The sum of the digits is therefore the sum of the digits of 390625, as all the other digits are zero.
\[ \text{Sum of digits} = 3 + 9 + 0 + 6 + 2 + 5 = 25 \] Step 4: Final Answer
The sum of the digits of the number \(625^{65} \times 128^{36}\) is 25.