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What is the greatest common factor of x and y ?

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Re: What is the greatest common factor of x and y ?  [#permalink]

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New post 18 Jun 2018, 20:06
longhaul123 wrote:
0 is also a multiple of every number and in this case if we consider x to be 0 then the GCD is going to 1 ? why shouldnt we consider x to be 0 here ??


Hello

In general when we talk about multiples/factors, we talk about non-zero positive integers only. This is how multiples/factors are defined in my opinion.
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Re: What is the greatest common factor of x and y ?  [#permalink]

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New post 27 Jun 2018, 22:36
did you mean to say that since 0 is a multiple of every integer then 0 can have any factors including 4 . Since the question is stressing on the point that it is just a multiple of 4 so 0 cant be considered.
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What is the greatest common factor of x and y ?  [#permalink]

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New post 12 Dec 2018, 00:52
[quote="stonecold"]A lot of solutions on this page already.
Here is mine
Clearly both 1 and 2 are alone insufficient.
combining them => x=4a y=4b and a=b+1 where b and b+1 will always be coprimes.


stonecold
how do you get a=b+1

If I understand it correctly,

Your solution is as follows:

S1) x=4a, y=4b
S2) x-y=4

S1) & S2) 4(x-y)=4 –> x-y=1 –> 4a-4b=1 –> a=b+1

Since a>b –> gcd(x,y)=gcd(4*(b+1), 4b)=gcd(4b+4-4b,4b)=gcd(4,4b)=4
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Re: What is the greatest common factor of x and y ?  [#permalink]

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New post 03 Jul 2020, 22:46
VeritasKarishma wrote:
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.


Hi VeritasKarishma

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 \(1\).
What can't we use this to arrive at E as the answer?

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Re: What is the greatest common factor of x and y ?   [#permalink] 03 Jul 2020, 22:46

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