GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 06 Aug 2020, 03:01 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If t is a positive integer and r is the remainder when t^2 + 5t + 6 is

Author Message
TAGS:

### Hide Tags

Intern  Joined: 20 Feb 2012
Posts: 13
If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

10
1
109 00:00

Difficulty:   85% (hard)

Question Stats: 55% (02:33) correct 45% (02:37) wrong based on 1473 sessions

### HideShow timer Statistics

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 -.-
Math Expert V
Joined: 02 Sep 2009
Posts: 65829
If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

19
39
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 remainders upon division by 7. Not sufficient.

Hope it's clear.
_________________
Director  G
Joined: 26 Oct 2016
Posts: 588
Location: United States
Schools: HBS '19
GMAT 1: 770 Q51 V44
GPA: 4
WE: Education (Education)
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

7
2
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.
_________________
Thanks & Regards,
Anaira Mitch
##### General Discussion
Board of Directors D
Joined: 01 Sep 2010
Posts: 3495
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

1
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
_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 65829
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

3
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.
_________________
Intern  Joined: 23 Dec 2012
Posts: 5
Concentration: General Management, Entrepreneurship
GPA: 3.5
WE: Consulting (Non-Profit and Government)
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

4
1
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:

 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

 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  and .

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.

Intern  Joined: 15 Aug 2014
Posts: 1
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

(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?
Manager  Status: PLAY HARD OR GO HOME
Joined: 25 Feb 2014
Posts: 122
Location: India
Concentration: General Management, Finance
Schools: Mannheim
GMAT 1: 560 Q46 V22
GPA: 3.1
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

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 _________________
ITS NOT OVER , UNTIL I WIN ! I CAN, AND I WILL .PERIOD.
Board of Directors D
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 5033
Location: India
GPA: 3.5
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

1
1
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)

_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
Manager  G
Joined: 21 Mar 2017
Posts: 133
Location: India
GMAT 1: 560 Q48 V20 WE: Other (Computer Software)
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

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.
_________________
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
When nothing seem to help, I would go and look at a Stonecutter hammering away at his rock perhaps a hundred time without as much as a crack showing in it.
Yet at the hundred and first blow it would split in two.
And I knew it was not that blow that did it, But all that had gone Before
.
Math Expert V
Joined: 02 Sep 2009
Posts: 65829
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

1
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.
_________________
Manager  B
Joined: 14 Nov 2018
Posts: 88
Location: United Arab Emirates
Concentration: Finance, Strategy
Schools: LBS '22 (I)
GMAT 1: 590 Q42 V30
GMAT 2: 670 Q46 V36 GPA: 2.6
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

Bunuel For st1) Why can we not replace (7p+6) for all values of t?
Math Expert V
Joined: 02 Sep 2009
Posts: 65829
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

gmatapprentice wrote:
Bunuel For st1) Why can we not replace (7p+6) for all values of t?

You CAN replace t with 7p + 6 in t^2 + 5t + 6. You'll get 7*something + 72. 7*something is divisible by 7, and 72 yields the remainder of 2 upon division by 7.
_________________
Manager  B
Joined: 14 Nov 2018
Posts: 88
Location: United Arab Emirates
Concentration: Finance, Strategy
Schools: LBS '22 (I)
GMAT 1: 590 Q42 V30
GMAT 2: 670 Q46 V36 GPA: 2.6
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

Bunuel You are right - I accidently took 129 instead of 119.

T^2+5t+6=(7p+6)^2+5(7p+6)+6
= 49p^2+84p+36+35p+30+6
= 49p^2+119p+72 ---

Many thanks!
Director  P
Joined: 21 Feb 2017
Posts: 678
GMAT 1: 690 Q45 V38
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

1) When t is divided by 7 --> $$t^2+5t+6$$.
First term $$t^2/7$$ gives remainder 6*6/7 =>, therefore, remainder 1,
second term $$5t/7$$ gives remainder as 2
and third term 6/7 as 6. adding 1+2+6/7 Remainder is 2. Sufficient

2) When t^2 is divided by 7, the remainder is 1 --> different values of t possible. Therefore insufficient.

Is this approach correct?
Manager  P
Status: wake up with a purpose
Joined: 24 Feb 2017
Posts: 195
Concentration: Accounting, Entrepreneurship
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

The patterns in remainder problems will always emerge fairly early when you plug in numbers. therefore, if you don't IMMEDIATELY realize a good theoretical way to do a remainder problem, you should get on the number plugging RIGHT AWAY.

I would IMMEDIATELY start plugging in numbers and using PATTERN RECOGNITION on a problem like this one. remainder problems usually show patterns after a very, very small number of plug-ins.

Statement (1):
it's easy to generate t's that do this: 6, 13, 20, 27,

... (note that 6 is a member of this list, and an awfully valuable one at that; it's quite easy to plug in)
try 6: 36 + 30 + 6 = 72; divide by 7 --> remainder 2
try 13: 169 + 65 + 6 = 240; divide by 7 --> remainder 2
try 20: 400 + 100 + 6 = 506; divide by 7 --> remainder 2
by this point i'd be convinced.

Note that 3 plug-ins is NOT good enough for a great many problems, esp. number properties problems.

however, as i said above, remainder problems don't keep secrets for long.
sufficient.

Statement (2):
it's harder to find t's that do this.
however, the gmat is nice to you. if examples are harder to find, then the results will usually come VERY quickly once you find those examples.
just take perfect squares, examine them, and see whether they give the requisite remainder upon division by 7.
the first two perfect squares that do so are 1^2 = 1 and 6^2 = 36.
if you don't recognize that (1^2)/7 gives remainder 1, then you'll have to dig up 6^2 = 36 and 8^2 = 64. that's not that much more work.
in any case, you'll have:

For 1 & 6:
1 + 5 + 6 = 12 --> divide by 7; remainder = 5
36 + 30 + 6 = 72 --> divide by 7, remainder = 2 (the work for this was already done above; you should NOT do it twice. i'm reproducing it here only for the sake of quick understanding.)

or

For 6 & 8:

36 + 30 + 6 = 72 --> divide by 7, remainder = 2 (the work for this was already done above; you should NOT do it twice. i'm reproducing it here only for the sake of quick understanding.)
64 +40 + 6 = 110 --> divide by 7, remainder = 5 either way, insufficient within the first two plug-ins!

Posted from my mobile device
_________________
If people are NOT laughing at your GOALS, your goals are SMALL.
Manager  B
Joined: 01 Apr 2020
Posts: 82
Location: India
GMAT 1: 650 Q46 V34 (Online) Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

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 remainders upon division by 7. Not sufficient.

Hope it's clear.

I think an easy way to prove (1) is sufficient is

t = 7k + 6 i.e. Remainder = 6 when t is divided by 7
Squaring both sides
$$t^2$$ = $$(7k)^2$$ + 2*7*6*k + 36
Remainder = 1 (as 7 * 5 = 35)

So now we have our "REMAINDERS" ready

and let's just put then in the equation

$$t^2+5t+6$$
Remainders = 1 + 6*5 + 6 = 1 + 30 + 6 = 37
Remainder = 2 (as 7 * 5 = 35)

Statement (2)
INSUFFICIENT as it is just SQUARED version of (1) and we don't know what the remainder can be when t is divided by 7.
GMAT Club Legend  V
Joined: 18 Aug 2017
Posts: 6511
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

[quote="dpark"]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.

given
t^2 + 5t + 6 is divided by 7
(t+3)*(t+2)= 7x+r
target find r
#1
When t is divided by 7, the remainder is 6.
t = 6,20,27
its observed that remainder is always 2
sufficient
#2
When t^2 is divided by 7, the remainder is 1.
t= 1 & 6
its observed that remainder is 5 & 2 insufficient
OPTION A ; sufficient
Intern  B
Joined: 19 Jun 2020
Posts: 9
Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  [#permalink]

### Show Tags

Statement 1 contains statement 2. However, statement 2 does not contain all the info of statement 1.

(1) t^2=49a+36=7b+1 (notice that we already have the info of statement 2)
5t=35a+30=7b+2
6/7 = 1 R6

9/2= 1 R2.
Since we can't infer the 5t equation from statement (2) the answer is A. Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is   [#permalink] 17 Jul 2020, 09:14

# If t is a positive integer and r is the remainder when t^2 + 5t + 6 is  