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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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23 Feb 2012, 15:21
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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.

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

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23 Feb 2012, 15:50
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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.

Hope it's clear.
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Re: Tricky? remainder question (gmatprep)  [#permalink]

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05 Jul 2012, 11:18
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.

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: Tricky? remainder question (gmatprep)  [#permalink]

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05 Jul 2012, 11:28
2
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.

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]

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08 Feb 2013, 10:37
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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.

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

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25 Aug 2014, 12:51
(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?
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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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06 Sep 2014, 05:17
1
Shehryar Khan wrote:
(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?

Hii Shehryar khan,
let me try to explain..

t^2/7=1----------given

as u can see,we have 2 values of 2..1^2/7=remainder 1 and 6^2/7=36/7=remainder 1..
when we substitute 1 in (t+2)(t+3)/7 ,we get remainder as 5..
when we substitute 6 in (t+2)(t+3)/7, we get remainder as 2..

So,since the results are inconsistent,we cant have one definite remainder,which we are required to find,as per the stem..so,not sufficient..

Please consider KUDOS if my post helped
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If t is a positive integer and r is the remainder when t^2+5  [#permalink]

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15 Oct 2016, 07:31
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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.

FROM STATEMENT - I INSUFFICIENT

The minimum value of t = 13

So, $$t^2+5t+6 = 13^2+5*13+6$$ = $$240$$

Now, $$\frac{240}{7}$$ = Remainder 2

FROM STATEMENT - II INSUFFICIENT

The minimum value of t = +1 and -1

So, $$t^2+5t+6 = 1^2+5*1+6$$ = $$12$$ Remainder 5

And $$t^2+5t+6 = -1^2+5*-1+6$$ = $$2$$ Remainder 2

Thus, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked, hence correct answer will be (A)

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

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04 Jan 2017, 04:14
I am sure that I am missing something because stem (1) tells that t=7q+6 so t=6,13,20,...
then we can try different values of t so that we can find out the reminder,
so it follows that (t+2)(t+3)=5p+r
for t=13 (13+2)(13+3)=240, 240/5 gives a reminder of 0
for t=20 (20+2)(20+3)=506, 506/5 gives a reminder of 1.
So stem (1) Not sufficient...
Please, can someone explain to me what I am missing? Thanks
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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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11 Feb 2017, 23:58
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First, solve the equation

t^2 + 5t + 6 = (t+2)(t+3)

statement (1)

t = 7q + 6
t can also be written as 7k-1, this means any multiple of 7 and then subtract 1 to the same when divided by 7 will always give a remainder 6

therefore,

(t+2)(t+3) = (7k-1+2)(7k-1+3) = (7k+1)(7k+2)

when (7k+1) will be divided by 7 the remainder will be 1
when (7k+2) will be divided by 7 the remainder will be 2

when the product of both is divided by 7 the remainder will be 2 for any value of k.
Sufficient.

Statement (2)
t^2 when divided by 7 the remainder is 1
We cannot follow the same strategy as above in this statement because here we are dealing with addition as compared to multiplication as statement (1)

this statement is even more simpler than the first one

t^2 + 5t + 6
t^2 = multiple of 7 + 1

t^2 + 6 = multiple of 7.

Therefore, we only have to find out if 5t/7 gives standard remainder across all cases.

5*5/7 = 35/7 = remainder 0
5*4/7 = 20/7 = remainder 6
5*6/7 = 30/7 = remainder 2.

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

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22 Aug 2017, 06:04
Hi Bunuel ,

As t^2 = 7m+1 so t^2 could be 1,8,15,22. Which gives the remainder as 5.
But you have taken the value as 1 and 6. Could you please explain why you have taken 6 instead of 8 in case 2.

When t^2 is divided by 7, the remainder is 1 --> different values of t possible: for example t=1 ort=6, which when substituted in (t+2)(t+3)(t+2)(t+3) will give different remainder upon division by 7.
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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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22 Aug 2017, 07:01
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Prashant10692 wrote:
Hi Bunuel ,

As t^2 = 7m+1 so t^2 could be 1,8,15,22. Which gives the remainder as 5.
But you have taken the value as 1 and 6. Could you please explain why you have taken 6 instead of 8 in case 2.

When t^2 is divided by 7, the remainder is 1 --> different values of t possible: for example t=1 ort=6, which when substituted in (t+2)(t+3)(t+2)(t+3) will give different remainder upon division by 7.

(2) reads: t^2 is divided by 7, the remainder is 1, not t.

t = 1 --> t^2 = 1 --> 1 divided by 7 gives the remainder of 1.
t = 6 --> t^2 = 36 --> 36 divided by 7 gives the remainder of 1.
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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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19 Apr 2018, 13:23
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 Bunuel

In terms of approach, can you clarify how one should look at the two statements? That is, your approach with st 1 is kind of algebraic and the latter was number plugging in. The former is a bit intuitive in this case (for me at least) but then how do you decide which approach befits statement 2 for instance?
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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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06 Sep 2018, 13:24
Abhishek009 wrote:
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.

FROM STATEMENT - I INSUFFICIENT

The minimum value of t = 13

So, $$t^2+5t+6 = 13^2+5*13+6$$ = $$240$$

Now, $$\frac{240}{7}$$ = Remainder 2

FROM STATEMENT - II INSUFFICIENT

The minimum value of t = +1 and -1

So, $$t^2+5t+6 = 1^2+5*1+6$$ = $$12$$ Remainder 5

And $$t^2+5t+6 = -1^2+5*-1+6$$ = $$2$$ Remainder 2

Thus, Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked, hence correct answer will be (A)

HiAbhishek009
Well why can't the min value of t=6

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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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06 Sep 2018, 13:44
VikramJS wrote:
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.

Hi Bunuel,

Can you explain how above has been used to arrive at answer. I think the user wanted to convey dividend in place of quotient

Thnaks
Probus
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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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06 Sep 2018, 14:56
Hichetan2u,

For this question is this approach correct

we can write the exp as (t+2)(t+3)

Stm1:$$\frac{t}{7}$$ gives reminder 6, then it must be that (t+1) will give reminder 0, and (t+2) &(t+3) will give reminder 1,2 respectively
then reminder is equal to 2

Sufficient

Stmt 2: $$t^2$$ divided by 7 gives reminder 1 , t could take value of 1 or 6 then exp gives different reminders.
Not Suff

My concern is regarding the approach used for Stmt 1. Is it correct

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

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08 Sep 2018, 01:35

I always thought the following to be true:

Here simplifying k^2 where k leaves a remainder of 1 with 7, the values of k can be 1 or 6; I thought we could apply the general rule that if 2 elements are multiplied, the remainder of the expression would be the remainder of the product of the individual remainders of each term of the expression.

For example, if we have an expression say 9k, and we know that k when divided by 7 is 1, then the following would be the remainder when 9k is divided by 7.

Remainder of 9 with 7 will be 2
Remainder of k with 7 will be 1

Multiplying remainders is 9k = 2 * 1 = 2
Thus the remainder of the expression 9k would be 2 when divided by 7

So extending that logic, if the expression is k^2, the remainder should be

1 * 1 = 1, thus the remainder would be 1. But clearly this is not the case as seen here, as 6 is also a valid answer.

I guess the core of my question is the above method that I used, is it outright wrong? Or is it applicable only in cases where expressions are of the form constant * variable like 9k and not when it is of the form variable * variable like k^2.
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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, 22:19
Bunuel

statement2:

t^2=7q+1

substituting in the given equation t^2+5t+6

(7q+1+5t+6)/7

(7q+5t+7)/7

(7(q+1)+5t)/7

5t+0 so remainder 0....

why don't we use this method?
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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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30 Sep 2018, 23:01
mahi816 wrote:
Bunuel

statement2:

t^2=7q+1

substituting in the given equation t^2+5t+6

(7q+1+5t+6)/7

(7q+5t+7)/7

(7(q+1)+5t)/7

5t+0 so remainder 0....

why don't we use this method?

It's not clear how you concluded that 5t will have the remainder of 0 when divided by 7. That's not correct.
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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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30 Sep 2018, 23:15
From statement 1 we have
t/7 the remainder is 6
t = 7k + 6
Going by the equation in question
t^2 = (7k+6)^2 = 49k^2+36+84k
Next,
t^2+5t = 49k^2+36+84k+5(7k+6) = 49k^2+36+84k+35k+30
Next,
t^2+5t+6= 49k^2+36+84k+35k+30+6 = 49k^2+36+84k+35k+36
Implies, t^2+5t+6 = 7(7k^2+12k+5k)+72 = 7(7k^2+17k)+7(10)+2
Implies, t^2+5t+6= 7(7k^2+17k+10)+2
This implies that the remainder r = 2.
So, Statement 1 is sufficient.
From statement 2 we have
t^2 =7x+1
Implies ,
t = (7x+1)^1/2
If the above result is substituted in the equation in the question stem then we get
t^2+5t+6 = 7x+1+5((7x+1)^1/2)+6 which cannot be simplified further elegantly.
Re: If t is a positive integer and r is the remainder when t^2+5   [#permalink] 30 Sep 2018, 23:15

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