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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] New post 23 Feb 2012, 14:21
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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.

[Reveal] Spoiler:
...
(1) means t=7n+6.
=> t²=(7n+6)²)=49n²+84n+36=49n²+84n+35+1
=> 5t=35n+30=35n+28+2

remainders 1+2+6=9=7+2

Remainder of t²+5t+6 is 2????

I dont really get this -.-
[Reveal] Spoiler: OA
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If t is a positive integer and r is the remainder when t^2+5 [#permalink] New post 23 Feb 2012, 14:50
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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.
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Re: Tricky? remainder question (gmatprep) [#permalink] New post 05 Jul 2012, 10:28
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carcass wrote:
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 yield different remainder upon division by 7. Not sufficient.

Answer: A.

Hope it's clear.



Sorry Bunuel but this part highlighted is not completely clear: on the left side of = we have the two roots of our equation; on the right (t+2) ----> is 7q+6+2 ---> 7q+8 (same reasoning for 7q+9) BUT why we equal this two part ?? Can you explain me please ??

Thank you


We are asked to find the remainder when t^2+5t+6 is divided by 7 or as t^2+5t+6=(t+2)(t+3), the remainder when (t+2)(t+3) is divided by 7.

Now, from (1) we have that t=7q+6. Substitute t with 7q+6 in (t+2)(t+3) to get (7q+8)(7q+9). So, finally we have that we need to find the remainder when (7q+8)(7q+9) is divided by 7 (t^2+5t+6=(t+2)(t+3)=(7q+8)(7q+9)).

Hope it's clear.
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Re: If t is a positive integer and r is the remainder when t^2+5 [#permalink] New post 08 Feb 2013, 09:37
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My approach:

From (1) - t/7 gives remainder = 6. So we can easily find remainder for t^2 and 5*t in the algebraic exp. t^2 + 5t + 6 using properties of remainder.

Therefore the remainder of algebraic exp. = 1 + 2 + 6 = 9 (7 + 2) = 2 ................. [A]


Properties of remainder:

[1] If a quotient is squared, the remainder also gets squared and we need to adjust the new remainder based on no. we get on squaring.

Here quotient = t, r = 6. So, t^2 will have r' = 36 (7*5 + 1). Therefore r' =1

[2] If a quotient is multiplied by an integer k , the remainder also gets multiplied and we need to adjust the new remainder based on no. we get on multiplication.

Here quotient = t, r = 6. So, 5t will have r'' = 30 (7*4 + 2). Therefore r'' = 2

We get [A] using [1] and [2].

Clearly Stmt (2) can have multiple possibilities i.e. if t= 1 (t^2 =1) and t =6 (t^2 = 36) both give remainder 1, so no unique answer and hence rejected.

Thus A is the answer.
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Re: Tricky? remainder question (gmatprep) [#permalink] New post 05 Jul 2012, 10:18
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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 yield different remainder upon division by 7. Not sufficient.

Answer: A.

Hope it's clear.



Sorry Bunuel but this part highlighted is not completely clear: on the left side of = we have the two roots of our equation; on the right (t+2) ----> is 7q+6+2 ---> 7q+8 (same reasoning for 7q+9) BUT why we equal this two part ?? Can you explain me please ??

Thank you
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Re: If t is a positive integer and r is the remainder when t^2+5 [#permalink] New post 05 Jul 2012, 14:58
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Re: If t is a positive integer and r is the remainder when t^2+5 [#permalink] New post 06 Nov 2013, 08:34
Wow. This is a tough problem. A is straightforward once you see it. B is time consuming if you take it all the way.
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If t is a positive integer and r is the remainder when t^2+5 [#permalink] New post 25 Aug 2014, 11:51
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(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.

Bunuel could you please explain why statement 2 is not sufficient and how the values for 't' could be either 1 or 6?
If t is a positive integer and r is the remainder when t^2+5   [#permalink] 25 Aug 2014, 11:51
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