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 \(1\).