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?

Question

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 the above, what is true?
A.if statement (1) by itself is sufficient to answer the question, but statement (2) by itself is not;
B. if statement (2) by itself is sufficient to answer the question, but statement (1) by itself is not;
C.if statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient;
D.if either statement by itself is sufficient to answer the question;
E.if statements (1) and (2) taken together are not sufficient to answer the question, requiring more data pertaining to the problem

Answer 1

First key to all data sufficiency questions is to find the "real" question.  Try rewriting the question and asking yourself "What info do I need to answer this?"

 

We can do reverse FOIL to rewrite the question as r=? when (t+2)*(t+3) / 7.  To answer this we need to know something about t.

 

Note that this is a value question rather than a Yes/No question.  For value questions, you need to arrive at one and only one answer.  If you can arrive at more than one answer the statement is insufficient.  Let's try to make these statements insufficient by testing cases.

 

Look at statement 1.  t could be 13, 20, etc.

(13+2)*(13+3)/7 = 15*16/7.  Can be written in prime factored form.  3*5*2*2*2*2/7 = 3*80/7 = 240/7.  210 is divisible by 7, so we just need to think of 30/7 which results in r = 2.

(20+2)*(20+3)/7 = 22*23/7.  Can be written in prime factored form.  2*11*23/7 = 46*11/7 = 506/7.  490 is divisible by 7, so we just need to think of 16/7 which results in r=2.

Sufficient

 

Look at statement 2.  t could be 6, 8, etc.

(6+2)*(6+3)/7 = 8*9/7 which results in r = 2

(8+2)*(8+3)/7 = 10*11/7 which results in r = 5

NOT Sufficient

 

If you're not sold on A alone, test one more number (like t = 27) to verify you keep getting r=2.  Third time's the charm.

 

Sure you can do it algebraically but learning to effectively test cases will benefit you more in the long run.

 

Answer 2

B is the answer. As you cycle values for t, 1 and 6 both satisfy statement 1 (as do 8 and 13, etc) but give different values for r so 1 alone is not sufficient. Statement 2 is is sufficient since 6, 13, 20 etc all satisfy statement 2 and all result in r = 2.