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If t is a positive integer and r is the remainder when t^2+5

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If t is a positive integer and r is the remainder when t^2+5  [#permalink]

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New post 30 Sep 2018, 22:28
dpark 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.



Statement (1) Says that when t is divided by 7, the reminder is 6.

Say : t = 7A + 6
t^2 = 47A^2 + 84A + 36
Hence, when t^2 is divided by 7, the remainder is 1.

5t = 35A + 30
Hence, when 5t is divided by 7, the remainder is 2.

Hence, the remainder of t^2+5t+6 is divided by 7 would be (1+2+6)/7 = 2

So, statement 1 is sufficient.

From statement 2, we cant derive what would be the value of t as t^2 can be 1 , 36, 64 ... so on. Hence, not sufficient.
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Re: If t is a positive integer and r is the remainder when t^2+5  [#permalink]

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New post 01 Nov 2018, 19:58
Bunuel 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?

First of all factor \(t^2+5t+6\) --> \(t^2+5t+6=(t+2)(t+3)\).

(1) When t is divided by 7, the remainder is 6 --> \(t=7q+6\) --> \((t+2)(t+3)=(7q+8)(7q+9)\). Now, no need to expand and multiply all the terms, just notice that when we expand all terms but the last one, which will be 8*9=72, will have 7 as a factor and 72 yields the remainder of 2 upon division by 7. Sufficient.

(2) When t^2 is divided by 7, the remainder is 1 --> different values of t possible: for example t=1 or t=6, which when substituted in \((t+2)(t+3)\) will give different remainder upon division by 7. Not sufficient.

Answer: A.

Hope it's clear.

Hi, I think in statement (2), t can not be equal to 1.
If we rephrase the statement, we get: t^2=7q+1, and t and q must be integer.
Here (7q+1) has to be a perfect square. So if t=1, the quantity (7q+1) becomes 8. We know 8 can not be equal to t^2 because t has to be an integer.
Correct me if I am wrong.

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Re: If t is a positive integer and r is the remainder when t^2+5 &nbs [#permalink] 01 Nov 2018, 19:58

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