What is the remainder when (\(67^{67}\) + 67) is divided by 68? GMAT Problem Solving

Question: What is the remainder when (67⁶⁷ + 67) is divided by 68?

A. 1
B. 11
C. 63
D. 66
E. 67

Answer: D

Approach Solution (1):
(xⁿ+ 1) will be divisible by (x+ 1) only whennis odd.
(67⁶⁷+ 1) will be divisible by (67 + 1)
(67⁶⁷+ 1) + 66, when divided by 68 will give 66 as remainder
Correct option: D

Approach Solution (2):
(67) = (-1) mod (68)
(67)⁶⁷ = (-1)⁶⁷ mod (68)
(67)⁶⁷ = (-1) mod (68)
Therefore,
(67)⁶⁷ + 67 = { (-1) + (67)} mod (68)
(67)⁶⁷ + 68 = (66) mod (68)
Therefore, the remainder is 66.
Correct option: D

Approach Solution (3):
We know that xⁿ +1 is divisible by x+1 when n is odd
So 67⁶⁷ + 1 is divisible by 67+1=68
So clearly we can see then if we add 66 on dividend then remainder must be 66 so
67⁶⁷ + 67 when divided by 68 the remainder is 66.
Correct option: D

“What is the remainder whenis divided by 68?”- 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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