swerve wrote:If t is a positive integer and r is the remainder when t^2 + 5t + 6 is divided by 7, what is the value of r?

(1) When t is divided by 7, the remainder is 6.
(2) When t^2 is divided by 7, the remainder is 1.

The OA is A.

Please, can anyone explain this DS question? I don't understand why A is the correct answer. I need help. Thanks.

Given: t is a positive integer and r is the remainder when t^2 + 5t + 6 is divided by 7

So, we have to find out the remainder of (t^2 + 5t + 6)/7 = the remainder of (t^2/7 + t/7 + 6/7)

Let's take each statement one by one.

(1) When t is divided by 7, the remainder is 6.

=> the remainder when t^2 is divided by 7 = the remainder when 6^2 is divided by 7 = the remainder when 36 is divided by 7 = 1

Thus, the remainder of t^2/7 + t/7 + 6/7 = the remainder of (1 +6 + 6)/7 = the remainder of (13)/7 = 6. Sufficient.

(2) When t^2 is divided by 7, the remainder is 1.

Case 1: Say t = 6, then the remainder when t^2 = 6^2 = 36 is divided by 7 is 1; and the remainder when t = 6 is divided by 7 is 6.

Thus, the remainder of t^2/7 + t/7 + 6/7 = the remainder of (1 +6 + 6)/7 = the remainder of (13)/7 = 6.

Case 2: Say t = 8, then the remainder when t^2 = 8^2 = 64 is divided by 7 is 1; and the remainder when t = 8 is divided by 7 is 1.

Thus, the remainder of t^2/7 + t/7 + 6/7 = the remainder of (1 +1 + 6)/7 = the remainder of (8)/7 = 1.

No unique answer. Insufficient.

The correct answer: A

Hope this helps!

-Jay
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