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22 Oct 2009, 20:04
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
56% (01:58) correct
44%
(01:30)
wrong
based on 835
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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) a is not divisible by 7
(2) a-b is divisible by 7
22 Oct 2009, 20:30
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mirzohidjon
(1) \(a=7p+r\) --> know nothing about b. Not sufficient.
(2) \(a-b=7q\) --> \(a+b=?\) --> Nor sufficient
(1)+(2) \(a=7p+r\) and \(a-b=7q\) --> \(b=a-7q\).
\(a+b=(7p+r)+(a-7q)=7p+r+7p+r-7q=7(2p-q)+2r\) --> \(7(2p-q)\) is divisible by 7. Since \(r\) is remainder from a being divided by 7, \(r\) and thus \(2r\) is not divisible by 7. So, \(a+b=7(2p-q)+2r\) is not divisible by 7. Sufficient.
Answer: C.
06 Apr 2012, 02:26
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alphastrike
Here is the logical explanation to the different answers (C and E) in the two cases (7 and 6).
First, let me explain why the answer is (C) in the original question.
Are the sum of integers a and b divisible by 7?
1) a is not divisible by 7
So a is of one of the following formats: 7n+1, 7n+2, 7n+3, 7n+4, 7n+5 and 7n+6
2) a-b is divisible by 7
The format of b is the same as the format of a i.e. if a = 7n + 3, then b = 7m + 3 since a and b vary by a multiple of 7
e.g. if a = 43 (= 7*6+1), then b could be 22 (= 7*3 + 1). The difference between them has to be a multiple of 7 so the remainder obtained by dividing a by 7 will be the same as the remainder obtained by dividing b by 7.
Using both statements, when you sum a and b, you get,
a+b = 7n+1 + 7m+1 = 7(n+m) + 2
or
a+b = 7n+2 + 7m+2 = 7(n+m) + 4
or
a+b = 7n+3 + 7m+3 = 7(n+m) + 6
etc
I hope you see that the remainder will be some even number (either 2 or 4 or 6 or 8 or 10 or 12). The remainder will never be 7 so the sum will never be a multiple of 7.
What happens if we replace 7 by 6?
Same logic.
a+b = 6n+1 + 6m+1 = 6(n+m) + 2
or
a+b = 6n+2 + 6m+2 = 6(n+m) + 4
or
a+b = 6n+3 + 6m+3 = 6(n+m) + 6
Let me stop here. Did you see something interesting? The remainder in the last case is 6 which means a+b is divisible by 6. In some cases, a+b will not be divisible by 6 and in some cases, it will be.
Hence your answer will be (E).
