If 10^{50} - 74 is Written as an Integer in base 10 Notation, What is the Sum GMAT Problem Solving

Question: If \(10^{50} - 74\) is written as an integer in base 10 notation, what is the sum of the digits in that integer?

1. 424
2. 433
3. 440
4. 449
5. 467

Answer:
Solution with Explanation:
Approach Solution (1):

Based 10 notations, or decimal notation, is just a way of writing a number using 10 digits: 1, 2, 3, 4, 5, 6, 7, 8, and 0 (usual way), in contrast, for example, to binary numeral system (base-2 number system) notation.

\(10^{50}\) has 51 digits: 1 followed by 50 zeroes;

\(10^{50} - 74\) has 50 digits: 48 9’s and 26 in the end;
So, the sum of the digit of \(10^{50} - 74\) equals to 48 * 9 + 2 + 6 = 440

Correct Option: C

Approach Solution (2):

\(10^3-74\)= 1,000 – 74 = 926 (1 nine)
\(10^4-74\)= 10,000 – 74 =9926 (2 nines)
\(10^5-74\)= 100,000 – 74 = 99926 (3 nines)
\(10^6-74\)= 1,000,000 = 999926 (4 nines)
.
.
.
In general, we can see that \(10^n-74\) will feature n – 2 9’s followed by 26
So, \(10^{50}-74\) will feature 48 9’s followed by 26

This means the sum of the digits = 48 (9) + 2 + 6 = 432 + 2 + 6 = 440

Correct Option: C

Approach Solution (3):

\(10^x-74\)--- last 2 digits are always 2, 6
\(10^2-74\)= 26
\(10^3-74\)= 926
\(10^4-74\)= 9926 and so on…
If x > 1,

The sum of the digits --- (x – 2) * 9 + 2 + 6
Hence, (50 – 2) * 9 + 8 = 440

Correct Option: C

“If\(10^{50}-74\)is written as an integer in base 10 notation, what is the sum of the digits in that integer?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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