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Retired Moderator P
Joined: 21 Aug 2013
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Re: What is the greatest common factor of x and y ?  [#permalink]

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

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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.
Senior Manager  S
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What is the greatest common factor of x and y ?  [#permalink]

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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,

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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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.

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

Warm Regards,
Pritishd
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Press +1 if this post helped you! Re: What is the greatest common factor of x and y ?   [#permalink] 03 Jul 2020, 22:46

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