Data Sufficiency

I haven't been able to get a response so I thought I might post here. Thank you for your help.


On the Numbers Properties Guide, there is the following Data Sufficiency Question:

9. Is the sum of integers A and B divisible by 7?
(1) a is not divisible by 7
(2) a-b is divisible by 7

1, not sufficient.
2, not sufficient because of the rule that if you add or subtract 2 non-multiples of an integer, the results could be a multiple of that integer or not.

but if we combine them, we know that A is not a multiple of 7. and A-B is a multiple of 7. So A (being a non-multiple) can only subtract B(another non-multiple) to get a multiple of 7, because of the rule the guide provided earlier, that if you add/subtract mulples of an integer, you get another multiple of that integer. but if you add/subtract a multiple with a non multiple, you get a non multiple.

so the answer I would imagine is E, a NON + NON = NON or Multiple of that integer.

but the guide provides the reason to why it is C, and it gives examples, but what is it that I am overlooking here? Why do the rules not apply to the way I tried to answer the problem?

Thanks,

Ray

Ah! Fairly tricky problem.

Try substituting various numbers to find a pattern. I'll try an algebraic solution.

(1) or (2) do not provide enough information alone.

Together,

From (1)

A = 7k1 + a ( a is the remainder and k1 some constant)

From (2)

A-B = 7k2

Sub (1) in (2)

(7k1+a) - B = 7k2

B = 7(k1-k2) + a

B = 7K + a

Now notice A and B leave the same remainder 'a'. 7 can never be written as sum to 2 equal integers.

That is, A+B = (7k1 + a) + (7K + a)

= 7(some constant) + 2a

7(some constant) is divisible by 7.

But, 2a is divisible by 7 only if a is divisible by 7.

Hence we can say A+B would not be divisible by 7. C

but where did i go wrong in my thought process when i was going through the problem?

This part is too generic for a generalization..

so the answer I would imagine is E, a NON + NON = NON or Multiple of that integer

Its not just some NON Multiple in both cases... its the same NON Multiple... see my explanation.. possible remainders for 7 would be..

1 - 1+1 =2
2- 2+2 = 4
3 - 3+3 =6
4 - 4+4 =8
5 - 5+5 = 10
6 - 6+6 = 12

Notice none of these numbers are divisible by 7. Makes sense?

fangtray wrote:but where did i go wrong in my thought process when i was going through the problem?

Q-(a+b)/7=?

stmnt 1(i.e. A is not a multiple of 7) is insuff, since there is no info about b

stmnt 2(i.e. (a-b)/7 ) is insuff. if a=14 b=7 the answ to the q-(a+b)/7 is yes. but if a=9 b=2 the answ is no

both stmnts-
since A cant be divided by 7 , then b cant be divided by 7 too to make (a-b) divisible by 7.

it means,that a+b is not divisible by 7

C is the answ

Great explanations guys! Well done.

LalaB wrote:Q-(a+b)/7=?

stmnt 1(i.e. A is not a multiple of 7) is insuff, since there is no info about b

stmnt 2(i.e. (a-b)/7 ) is insuff. if a=14 b=7 the answ to the q-(a+b)/7 is yes. but if a=9 b=2 the answ is no

both stmnts-
since A cant be divided by 7 , then b cant be divided by 7 too to make (a-b) divisible by 7.

it means,that a+b is not divisible by 7

C is the answ

I agree with your statement since A can't be divided by 7, B can't be divided by 7 to make A-B divisible by 7.

So A and B are both Non-mulitples of 7.

how about 15 + 6 =21? in this case, A + B would be divisible by 7. This is what i'm getting at. I understand that this equation i gave as an example does not satisfy A-B = something divisble by 7, but i'm trying to use MGMAT's rules of if A-B is divible by 7, and A is not divisible by 7, then B must not be divisible by 7. and if A and B are both NON MUltiples of 7, than they can be added to be divisible, or not..

fangtray wrote:I haven't been able to get a response so I thought I might post here. Thank you for your help.


On the Numbers Properties Guide, there is the following Data Sufficiency Question:

9. Is the sum of integers A and B divisible by 7?
(1) a is not divisible by 7
(2) a-b is divisible by 7

1, not sufficient.
2, not sufficient because of the rule that if you add or subtract 2 non-multiples of an integer, the results could be a multiple of that integer or not.

but if we combine them, we know that A is not a multiple of 7. and A-B is a multiple of 7. So A (being a non-multiple) can only subtract B(another non-multiple) to get a multiple of 7, because of the rule the guide provided earlier, that if you add/subtract mulples of an integer, you get another multiple of that integer. but if you add/subtract a multiple with a non multiple, you get a non multiple.

so the answer I would imagine is E, a NON + NON = NON or Multiple of that integer.

but the guide provides the reason to why it is C, and it gives examples, but what is it that I am overlooking here? Why do the rules not apply to the way I tried to answer the problem?

Thanks,

Ray

I would plug in numbers, but here's another approach:

Statement 1: a is not a multiple of 7
No information about b.
INSUFFICIENT.

Statement 2: a-b is a multiple of 7
Let's determine what must be true for both a-b and a+b to be multiples of 7.
If a+b is a multiple of 7 and a-b is a multiple of 7, then the sum of (a+b) and (a-b) is a multiple of 7:
(a+b) + (a-b) = multiple of 7
2a = multiple of 7
a = multiple of 7.
Thus, if a is NOT a multiple of 7, then a+b is NOT a multiple of 7.

If a IS a multiple of 7, then -- since a-b is a multiple of 7 -- we know that b also is a multiple of 7.
Thus, a+b = multiple of 7 + multiple of 7 = multiple of 7.
INSUFFICIENT.

Statements 1 and 2:
As noted above, if a is not a multiple of 7 (statement 1), then in statement 2, a+b is not a multiple of 7.
SUFFICIENT.

The correct answer is C.

Simple way of Doing

Statement 1:
b not known..Insufficient

Statement 2:
a-b=7
14-7=7 ...(7 factors of a,b) . a+b=21..divisible by 7
15-8=7 ...(7 not factors of a,b ). a+b=23...not divisible by 7

Insufficient

Combining(1) and (2)
"a" is not a divisible by 7 and (a-b) is divisible by 7 simply says that b is also not divisible by 7(by 2nd part of Statement 2)

so, (a+b) not divisible by 7
(C)

yes absolutely right!!! Even I got the answer as C.
The reason is that when I am sure that (a+b) is never going to be divisible by 7, Hence C.

It is mentioned in Statement 1 that a is not divisible by 7
Statement 2 says that(a-b) is divisible by 7.
So definately b is also not going to be divisible by 7.

Hence addition of both non-divisible numbers will be a non- divisible number in itself...

I hope this explanation helped...