Question

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

• A. \(21\)
• B. \(221\)
• C. \(324\)
• D. \(255\)

Hint:

When dealing with expressions like $10^n + 10^m - k$, avoid direct expansion. Instead:
Treat powers of 10 as place-value blocks (leading 1 followed by zeros).
Rewrite as $10^n + (10^m - k)$ and analyze the smaller block $10^m - k$.
Use patterns like $10^n - 1 = \underbrace{99\ldots9}_{n\text{ times}}$ and adjust for the subtraction. This makes digit-sum problems much faster and cleaner.

Answer 1

The Correct Option is B

Step 1: Understand the structure of the number. First, note: 

10⁵⁰ = 1 followed by 50 zeros, 
10²⁵ = 1 followed by 25 zeros.

So,

10⁵⁰ + 10²⁵ is a 51-digit number with:
- a 1 in the 10⁵⁰ place,
- a 1 in the 10²⁵ place,
- all other digits 0.
 

Thus, it looks like:

10⁵⁰ + 10²⁵ = 1 followed by 25 zeros, then 1 followed by 24 zeros.

Step 2: Rewrite the given expression. We can group as:

10⁵⁰ + 10²⁵ - 123 = 10⁵⁰ + (10²⁵ - 123).

Focus on:

10²⁵ - 123.

Step 3: Express 10²⁵ - 123 in digit form. Observe the pattern:

10³ - 123 = 877, 
10⁴ - 123 = 9877, 
10⁵ - 123 = 99877.

In general:

10ⁿ - 123 = a number with (n-3) nines followed by 877.

So,

10²⁵ - 123 = 22 nines followed by 877.

This is a 25-digit number: 22 nines followed by 877. Step 4: Combine with 10⁵⁰. Now:

10⁵⁰ + (10²⁵ - 123)

has:

- the leading part from 10⁵⁰: a 1 followed by 25 zeros,
- then the 25-digit block 10²⁵ - 123 = 22 nines followed by 877.

So the full number is:

1 followed by 25 zeros, then 22 nines followed by 877.

Step 5: Sum of digits. Digits:

- One leading digit 1,
- 25 zeros,
- 22 nines,
- digits 8, 7, 7.

Sum of digits:

1 + (25 * 0) + (22 * 9) + 8 + 7 + 7 = 1 + 198 + 8 + 7 + 7 = 1 + 198 + 22 = 221.

So, the required sum of digits is:

221.