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If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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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. ... (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 + 5t + 6 is
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23 Feb 2012, 14:50
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. Answer: A. Hope it's clear.
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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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11 Feb 2017, 22:58
First, solve the equation t^2 + 5t + 6 = (t+2)(t+3) statement (1) t = 7q + 6 t can also be written as 7k1, 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) = (7k1+2)(7k1+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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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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05 Jul 2012, 10:18
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.
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 + 5t + 6 is
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05 Jul 2012, 10:28
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 + 5t + 6 is
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08 Feb 2013, 09:37
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: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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25 Aug 2014, 11:51
(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 + 5t + 6 is
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06 Sep 2014, 04:17
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=1given 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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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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15 Oct 2016, 06:31
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 INSUFFICIENTThe 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 INSUFFICIENTThe 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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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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22 Aug 2017, 05:04
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 or t=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 + 5t + 6 is
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22 Aug 2017, 06:01
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 or t=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 + 5t + 6 is
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24 Dec 2019, 09:19
Bunuel For st1) Why can we not replace (7p+6) for all values of t?



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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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24 Dec 2019, 09:44
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.
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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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24 Dec 2019, 10:42
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!



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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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25 Jan 2020, 04:12
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.
Hence answer is A.
Is this approach correct?



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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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06 Mar 2020, 12:25
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 plugins.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 plugins 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 plugins! answer (a) Posted from my mobile device
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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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20 May 2020, 14:57
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.
Answer: A.
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.



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Re: If t is a positive integer and r is the remainder when t^2 + 5t + 6 is
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11 Jul 2020, 07:32
[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



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




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