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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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30 Sep 2018, 23:28
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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01 Nov 2018, 20:58
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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 give different remainder upon division by 7. Not sufficient.

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  [#permalink]

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20 Feb 2019, 14:19
Let me give an algebric approach to solve second part.

You need to get the value of remainder when t^2+5t+6 is divided by 7.

Since, t^2 leaves a remainder 1 on division with 7. t^2+6 will leave a remainder of 0.
Or
t^2+6 will be completely divisible by 7.
Now what you need to find is remainder when 5t is divided by 7.

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

We can say that here t^2-1 will be divisible by 7. Or to say (t-1)*(t+1) will be divisible by 7.

Please note that t-1,t,t+1 would be consecutive numbers. hence if (t-1)*(t+1) is divided by 7 then either t-1 OR t+1 will be completely divisible by 7.

( Both can not be completely divisible as the gap between them has to be atleast 7 to be completely divisible).

In either case that is if t-1 is completely divisible the remainder on dividing t by 7 would be 1.

OR if t+1 is divisible by 7, the remainder on dividing t with 7 would be 6.

Hence we can have 2 remaindes for t i.e 1 OR 6.

5t on dividing with 7 will leave either 5 or (30/7=2) as remainder . Hence the possible values for remainder is 2 or 5.

This is clearly insufficient .

This thought process is long but illustrates why 2 and 5 are logical values for remainder once we substitute.

Please hit Kudos if you like the answer.

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Re: If t is a positive integer and r is the remainder when t^2+5   [#permalink] 20 Feb 2019, 14:19

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