When a Number A is Divided by 6, the Remainder is 3 and When Another GMAT Problem Solving

Question: When a number A is divided by 6, the remainder is 3 and when another number B is divided by 12, the remainder is 9. What is the remainder when A^2+B^2 is divided by 12?

1. 4
2. 5
3. 6
4. 10
5. Cannot be determined

“When a number A is divided by 6, the remainder is 3 and when another”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide”. 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:
When a number A is divided by 6, the remainder is 3
In other words, A is 3 greater than some multiple of 6.
In other words, A = 6k + 3, for some integer k

When B is divided by 12, the remainder is 9.
In other words, B is 9 greater than some multiple of 12
In other words, B = 12j + 9, for some integer j

What is the remainder when A² + B² is divided by 12?
We have: A² + B² = (6k + 3)² + (12j + 9)²
Expand and simplify: A² + B² = (36k² + 36k + 9) + (144j² + 216j + 81)
Simplify: A² + B² = 36k² + 36k + 144j² + 216j + 90
Rewrite 90 as follows to get: A² + B² = 36k² + 36k + 144j² + 216j + 84 + 6
Factor out at 12 from the first five terms to get: A² + B² = 12(3k² + 3k + 12j² + 18j + 7) + 6

We can now see that A² + B² is 6 greater than some multiple of 12.
So, when we divide A² + B² by 12, the remainder will be 6

Correct Answer: C

Approach Solution 2:
We can write A as A = 6i + 3 where i is any nonnegative integer. Similarly B = 12j + 9 (Note: do not use i here as A and B are unrelated). Next we can write A2 + B2 as:

(6i+3)2+(12j+9)2

Now this next step is important, we are trying to find the remainder after dividing by 12. We can immediately eliminate any term that is a multiple of 12 since those terms will be divisible by 12, and do not affect the remainder. Whatever left that is not divisible by 12 will be part of our remainder.

To demonstrate the first step: (12j+9)2 will give us three terms but we already know two of them are multiples of 12. The only useful term we will keep is 9^2 = 81.

Thus (6i+3)^2 + (12j+9)^2 => 36i^2 + 2 ∗ 6i ∗ 3 + 9 + 81 => 9 + 81 => 90 => 6.

90/12 gives us a remainder of 6.

Correct Answer: C

Approach Solution 3:
Given: when a number A is divided by 6, the remainder is 3 and when another number B is divided by 12, the remainder is 9.

Asked: What is the remainder when A^2+B^2 is divided by 12?

A = 6k + 3; where k is an integer
B = 12m + 9; where m is an integer

A^2 + B^2 = (6k + 3)^2 + (12m + 9)^2 = 36k^2 + 36k + 9 + 144m^2 + 216m + 81 = (36k^2 + 36k + 144m^2 + 216m) + 90 = (36k^2 + 36k + 144m^2 + 216m + 84) + 6 = 12K + 6

Remainder when A^2+B^2 is divided by 12 = 6

Correct Answer: C

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