What is the Remainder When\( 2^{99}\) is Divided by 99? GMAT Problem Solving

Question: What is the remainder when \(2^{99}\) is divided by 99?

1. 17
2. 15
3. 13
4. 11
5. 9

Correct Answer: A

Approach Solution (1):

Having said that, you can ease it logically by seeing that all choices are 17 or less, so when you divide by 33 too, the remainder should be same.

Now, \(2^{99}=(2^5)^{19}*2^4=32^{19}*2^4\)

Now 32 will leave -1 as remainder

So remainder = \((-1)^{19}*16=-1*16=-16,or 33-16=17\)

Approach Solution (2):

What will be the remainder when \(2^{99}\) is divided by 9, a factor of 99.

\(2^{99}=(2^3)^{33}=8^{33}\)

8 will leave a remainder 1, so R = \((-1)^{33}=-1 or 9-1=8\)

Since we are looking for remainder with 99, remainder can be anything in the form 9k + 8

So the answer will be one of 8, 17, 23…

Only 17 is there so our answer must be 17

A point on one of the solution that since 99 = 3 * 3 * 11, so remainder = 3 + 3 + 11 is wrong

The solution doesn’t even talk of \(2^{99}\), so anything \(2^3or2^{13}or5^1\) and so on will give 17 as an answer.

“What is the remainder when \(2^{99}\) is divided by 99?”- 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.

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