CAT 2025

Slot 3

QA

Question & SolutionQues & Sol

The sum of all the digits of the number $(10^{50}+10^{25}-123)$, is

1. Concept Used

• Topic: Number System — Powers of 10 and Sum of Digits

• Concept:

The number (10^n-123) follows a predictable digit pattern:

$$ 10^3-123=877 $$

$$ 10^4-123=9877 $$

$$ 10^5-123=99877 $$

$$ 10^6-123=999877 $$

Thus, for (10^n-123), there are (2) digits equal to (7), (1) digit equal to (8), and ((n-3)) digits equal to (9).

2. Calculation

The given expression can be written as:

$$ 10^{50}+10^{25}-123 $$

$$ =10^{50}+(10^{25}-123) $$

For (10^{25}-123), there are:

• (2) digits of (7)
• (1) digit of (8)
• (25-3=22) digits of (9)

When this number is added to (10^{50}), the newly introduced digits are zeros, except for the leftmost digit, which contributes (1) to the digit sum.

Therefore, the sum of the digits is:

$$ (7\times2)+(8\times1)+(9\times22)+(0\times25)+1 $$

$$ =14+8+198+0+1 $$

$$ =221 $$

3. Solution

Answer = Option B ✅

The sum of the digits of (10^{50}+10^{25}-123) is 221.