maryann wrote:
What is the greatest common factor of x and y
1. x and y are both divisible by 4
2. x - y = 4
Stmnt 1: We know now that 4 is a factor of both. But is it the highest common factor, we do not know yet. There could be another factor common between x and y and hence highest common factor could be greater than 4. e.g. 4 and 16 have 4 as highest common factor but 12 and 36 have 12 as the highest common factor though both pairs have 4 as a common factor.
Stmnt 2: We know that x and y differ by 4. So they could have any of 1/2/4 as their highest common factor (Explanation given below) e.g. 7 and 11 have 1 as common factor while 2 and 6 have 2 as greatest common factor.
Taking both together: From stmnt 1, x and y have 4 as a common factor. From stmnt 2, x and y have one of 1/2/4 as highest common factor. Hence 4 is the highest common factor.
Answer (C).
Explanation:
Notice a few things about integers:
-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16......
Every number is a multiple of 1
Every second number is a multiple of 2
Every third number is a multiple of 3
Every fourth number is a multiple of 4 and so on...
If I pick any 2 consecutive integers, one and only one of them will be a multiple of 2: e.g. I pick 4, 5 (4 is a multiple of 2) or I pick 11, 12 (12 is a multiple of 2) etc..
If I pick any 3 consecutive integers, one and only one of them will be a multiple of 3: e.g. I pick 4, 5, 6 (6 is a multiple of 3) or I pick 11, 12, 13 (12 is a multiple of 3) etc..
This means that if I pick any two consecutive integers, they will have no common factor other than 1. (Say if 5 was their common factor, the numbers would be at least 5 apart e.g. 5 and 10.They cannot be consecutive. If 11 was their common factor, the numbers would be at least 11 apart e.g. 11 and 22. They cannot be consecutive. etc)
If I pick two integers with a difference of 4 between them, the only common factors (other than 1) they can have are 2 and/or 4
e.g. 2 and 6 have 2 as a common factor. 4 and 8 have 2 and 4 as common factors.
I guess I am missing out on something very fundamental, but unable to figure out. Any help would be great.
If we take \(x = 4\) and \(y = 0\) then both \(x\) and \(y\) are divisible by \(4\) and both numbers are \(4\) units apart, hence these satisfy both the statements and give GCD of \(1\).
Cheers. Wishing Luck to Every GMAT Aspirant
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