For an added challenge, time yourself to 4 minutes tops to

1. D?
2. B? (B because x value of l and k we know from the question itself - the value is x=4 for both l and k - am not sure if this logic is correct).

For the first question, it is helpful to factor, as it is for most questions involving number properties

First question

ST. 1
7n+6= t
let n=2
t= 20
then t^2+5t+6= 506 which yields remainder of 2 when divided by 7

try with n=3, remainder will be 2 again
SUFF

ST. 2
7n+1= t^2
let n= 5
then t= 6
and t^2+5t+6= 72 which leaves remainder of 2 when divided by 7
SUFF

D

Wrong.

I said D as well.

OA is A.





Like I said, Hard Quant Q's

First question:

I. Factor t^2+5t+6 into (t +3)(t + 2). If t/7 has remainder 6, then t = k*7+6, where k is an integer. So t+3 = k*7 + 9 = (k+1)*7 + 2.
t+2 = (k+1)*7 + 1. Multiplying them gives you:

(k*7)^2*49 + (k+1)*21 + 2. Whatever t is, the remainder when t^2+5t + 6 is divided by 7 will be 2. SUFF.

II. t^2/7 has a remainder of 1. Then t^2+6 is divisible by 7. So the value of r is the remainder when 5t is divided by 7. We don't have enough information to know. Since t^2/7 has a remainder of 1, t could be 6 (36-1 = 35), in which case r = 2. (30/7). Or t could be 8 (64- 1 = 63), in which case r = 5 (40/7).

So it's A.

John, fantastic explanation. I understand how to do the question now. Thanks I have to brush up on my remainder arithmetic.

I think the explanation provided by r019h was correct for first statement.

But for the second statement, the remainder is not 2, when some other number is considered, for e.g. when n = 9, in statement 2, the remainder is not 2. So the answer is not D, but it is A.

Honestly, I think the approach taken by r019h was much more straightforward and can be quick.

I dont think this is that hard...



asks what is r if (t+3)(t+2) is divided by 7!

1) says t=7N+6

so (t+3)=7N+9 => 7(N+1)+2
(t+2) => 7N+8 => 7(N+1)+1

so you will notice that r will always be 2... for any value of N...

sufficient

2) t^2=7N+1

t=6 then in that case (9)(8)/7 gives remainder 2...

t can also be 8... in that case (10)(11)/7 gives remainder 5..insuff

A it is..

excellent approach..

agree with A.

can be done this way as well:

1: since t has 6 reminder if t is divided by 7, t = 7k +6.

= t^2 + 5t + 6
= (7k+6)(7k+6) + 5 (7k+6) + 6
= 49k^2 + 42k + 42k + 36 + 35k + 30 + 6
= 49k^2 + 119k + 72

in the above expression, 49k^2 and 119k are evenly divided by 7. so remains 72 which as 2 as reminder when it is divided by 7.

so suff...

2: t^2 has 1 reminder if t^2 is divided by 7.

t could be 6 or 8 or 15 each has 1 as reminder so the reminder of the expression t^2 + 5t + 6 is different with t values. nsf.

A.

All these are great explanations guys!!! Greatly appreciated.

If i may add my thoughts...i thought the best way to solve it, since i am nowhere near as advanced as all of you are is to plug in nuymbers.

I tried different ones, but only 13 diveded by 7 gave the reminder 6. All other numbers will divide and give reminders but not 6. Since t is 13 if we do the brutal multiplications and divisions we could find the reminder of the main quadratic equation. Therefore SUFF.

The second one states that t*2 divided by 2, reminder is 1. Now off the top of my head i know that t can be either positive or negative, there may be differentmunbers giving the same remainder 1, thefore i ignore it and call INSUFF.
Thereofre i would go with A

Please let me know what you guys think about my reasoning.

I think these approaches are too time consuming. I just finished my GMAT today and I saw a problem VERY VERY similar to this question. The general idea/concept was the same, but the quadratic and divisor was different. Here's my take on how to approach (1):

The quadratic factors to: (t + 2)(t +3)

hmm...what pattern have we consistently seen involving divisors and quadratics? I'm thinking consecutive integers.

(1) Provides info on t so add to this consecutive integer string: t, t + 1, t + 2, t + 3

we know t has a remainder 6 when divided by 7 so t + 1 MUST be divisible by 7 which means t + 2 will have a remainder of 1 when divided by 7 and t + 3 will have a remainder of 2 when divided by 7. How do I know this? Simple: remainders increase from 0 to x - 1 as you iterate over consecutive integers and reset back to 0 once you reach a number that divides evenly. 7%7 = 0, 8%7 = 1, 9%7 = 2 etc etc where % is the modulus function (gives the remainder).

So now pick two values for t + 2 and t + 3 which have remainders 1 and 2, respectively.
8 and 9: 8*9 = 72; 72%7 = 2
15 and 16: 15*16 = 240%7 = 2
etc etc

SUFF. This method is simple, easy to understand and just involves picking numbers - no long winded quadratic calculations necessary.

(2) I'm not going to discuss because I just wanted to mention how I solved for (1)