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

Let's solve the problem step-by-step:

The given expression is \(10^{50} + 10^{25} - 123\). 

1. First, let's understand \(10^{50}\) and \(10^{25}\):
   • \(10^{50}\) is a number represented as 1 followed by 50 zeros: \(100\ldots0\) (50 zeros).
   • \(10^{25}\) is a number represented as 1 followed by 25 zeros: \(100\ldots0\) (25 zeros).
2. The sum \(10^{50} + 10^{25}\) results in a number with a 1 followed by 25 zeros, a 1, and then 24 more zeros:
   • This number is \(1000\ldots0001000\ldots00\), where the first 1 is at the 50th place and the second 1 is at the 25th place.
3. Now, we subtract 123 from this number:
   • The number with 1 at the 50th place and the 1 at the 25th place appears as \(1000000000000000000000001000000000000000000000000 - 123\).
   • When 123 is subtracted, it affects the last three digits of this large number. The impact will be at the end of calculations affecting the digits in those places.
4. To find the sum of the digits, consider:
   • The digits derived from \(10^{50} + 10^{25} - 123\) should be manually calculated in parts.
   • The position adjustments will result in a pattern that keeps all digits zero except a sequence of 999 from subtraction influencing the terminal digits.
5. Final Steps:
   • Calculate sum of the resulting digits directly.

After these steps, the calculated sum of all the digits of the number is \(221\).

Thus, the correct answer is \(221\).

Answer 2

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.