What is the Remainder when 51^203 is Divided by 7? GMAT Problem Solving

Question: What is the remainder when 51^203 is divided by 7?

(A) 4
(B) 2
(C) 1
(D) 6
(E) 5

“What is the remainder when 51^203 is divided by 7?” - is a topic of the GMAT Quantitative reasoning section of GMAT. 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.

Solution and Explanation:

Approach Solution 1:

Given that 51 ^ 203 is divided by 7 and it has asked to find out the remainder.

Since 51/7 = 7 R 2, the remainder when 51^203 is divided by 7 is the same as when 2^203 is divided by 7.

Now notice that 2^203 = (2^3)^67 x 2^2 = 8^67 x 4. Since 8/7 = 1 R 1, the remainder when 8^67 is divided by 7 is the same as when 1^67 is divided by 7. Furthermore, the remainder when 8^67 x 4 is divided by 7 is the same as when 1^67 x 4 is divided by 7. Since 1^67 x 4 = 4 and 4/7 = 0 R 4, the remainder is 4.

A is the correct answer.

Correct Answer: A

Approach Solution 2:

Given that 51^203 is divided by 7 and it has asked to find out the remainder.

51^203 mod 7= 2^203 mod 7
2^203=2^(3∗67+2) = (2^3)^67 . 2^2

2^203 mod 7
= (2^3) ^ 67. 2^2 mod 7

= 8^67 * 4 mod 7
= 4 mod 7

Hence, the remainder will be 4 in this case.

Correct Answer: A

Approach Solution 3:

Given that 51 ^ 203 is divided by 7 and it has asked to find out the remainder.

Every 51 after dividing by 7 leaves a remainder of (2).
Thus, 203 51s will leave 203 (2) as remainders.
So, 51^203 / 7 -> (2) ^ 203 / 7
Group three 2s to obtain an 8. On dividing this by 7, you have 1 as the remainder.
67 such groups would be created, and two 2s would be left ungrouped.
Thus (2)^203/7 ---> (8^67 ) x (2) x (2)
->(4)/7 = +4 as the remainder.

Hence, 4 is the remainder, and option A is the correct answer.

Correct Answer: A

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