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Question on modulo arithmetic

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by OneTwoThreeFour » Sat Apr 23, 2011 10:37 am
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.

My question is primarily on statement one. According to modulo arithmetic, the remainder for statement 1 is 2, since
remainder of t^2 is 36/7=1, 5t is 30/7=2, and remainder of 6/7 is 6. Thus 1+2+6=9 and 9-7=2. So the value of r is 2.

The proof for remainder of 5t/7 is given by:

t/y=Q + r/y (Where y is the divisor, r is the remainder, and Q is the quotient.)
5 * (t/y) = 5Q + 5r/y
This works, but my proof for t^2/7 is:
t * (t/y)= tQ + t*r/y. (t*r/y cannot equal to t^2/y for all circumstances.)
So what I am doing wrong? I know that the remainder for t^2/y is the same as r^2/y, so what is a correct proof for t^2/y?

Secondly, is there a quicker way to solve statement 2 from a conceptual approach instead of picking numbers?

Thanks!
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by OneTwoThreeFour » Sat Apr 23, 2011 10:40 am
Sorry, this belongs in the ds section. How do I delete posts in the new beatthegmat?
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by MAAJ » Sun Apr 24, 2011 8:29 am
IMO the answer should be [spoiler](A)[/spoiler]

Here's what I did:

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

Factorize: t^2 + 5t + 6 to get (t + 3)(t + 2)
We can multiply and add remainders as long as we correct the excess.
Because the remainder of t divided by 7 is 6 then we can deduce that the remainder of
(t + 3) must be 6 + 3 = 9 correcting the excess we get remainder = 2 and that the remainder of (t + 2) must be 6 + 2 = 8 correcting the excess we get remainder = 1

Now we multiply both remainders to get r = 2, in this case there is no excess over 7. This statement is sufficient.

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

t * t = some # that when divided by 7 the remainder is 1. If we look up for numbers (yeah... had to pick numbers)that satisfy this, we find that t could be 1, 6 or 8:

1 * 1 = 1
6 * 6 = 36
8 * 8 = 64

If we are able to find the remainders of each term independently, then we will be able to find the remainder of the equation. So, from t^2 + 5t + 6, we know that:

- The remainder of (t^2)/7 is 1
- The remainder of 6/7 must be 6
- The remainder of (5/7) must be 5
- The remainder of (t/7) could be 1 OR 6 (If 6/7 r = 6)

Hence the remainder of (5t/7) could be r = 5*1 = 5
Or could be 5 * 6 = 30 if we correct the excess we get r = 2

Therefore, the total remainder of the equation could be 1 + 5 + 6 = 12 -> r = 5
Or it could be 1 + 2 + 6 = 9 -> r = 2

So this statement is insufficient.
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