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# Which of the following is the greatest possible common divisor of two

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Re: Which of the following is the greatest possible common divisor of two [#permalink]
Amazing question.
GCD can almost be equal to smaller of the two integers.
TO maximise that=>
higher number => 122
Smaller number => 61

Hence Max GCD=> 61

Smash that E.
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Re: Which of the following is the greatest possible common divisor of two [#permalink]
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GCD of two different integers is a factor of both of them, first of all. So whatever two integers we choose less than 124 - both have to be multiples of that GCD.
Lets go by options here.

A) 123.. but next multiple of 123 will be >124. So not possible
B) 122. same as above. not possible
C) 63. Again same. next multiple of 63 is 126
D) 62. Again next multiple of 62 is 124. But both numbers must be less than 124

This leaves only option E, 61 as the correct answer
(61 and 122 have a GCD of 61)
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Re: Which of the following is the greatest possible common divisor of two [#permalink]
chetan2u: i am unable to understand the reasoning behing it. Why it can't be 123?
Could you please help me here?

Regards.
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Re: Which of the following is the greatest possible common divisor of two [#permalink]
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Expert Reply
kaulmeankit08 wrote:
chetan2u: i am unable to understand the reasoning behing it. Why it can't be 123?
Could you please help me here?

Regards.

Our answer has to be greatest possible common divisor of TWO numbers less than 124.
Now if it is 123, then WHICH are the two numbers which can be divisible by 123. It us just 1, that is 123.
Similarly for 122..

So 122 and 61 are divisible by 61as 122 is 61*2..
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Re: Which of the following is the greatest possible common divisor of two [#permalink]
chetan2u wrote:
kaulmeankit08 wrote:
chetan2u: i am unable to understand the reasoning behing it. Why it can't be 123?
Could you please help me here?

Regards.

Our answer has to be greatest possible common divisor of TWO numbers less than 124.
Now if it is 123, then WHICH are the two numbers which can be divisible by 123. It us just 1, that is 123.
Similarly for 122..

So 122 and 61 are divisible by 61as 122 is 61*2..

Now I got it. thank you for putting it in simple terms
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Re: Which of the following is the greatest possible common divisor of two [#permalink]
Expert Reply
Bunuel wrote:
Which of the following is the greatest possible common divisor of two different positive integers, both smaller than 124?

A. 123.
B. 122.
C. 63.
D. 62.
E. 61.

Let x be the greatest possible common divisor of two distinct positive integers, each less than 124. To keep x as large as possible, we need to have the factors of the two integers besides x to be as small as possible. Thus, we need to take the two integers to be x and 2x. Now, since 2x is the greater of these two integers, we should let 2x to be the greatest even number less than 124, which is 122. Then, 2x = 122 and therefore, x = 61.

Answer: E
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Re: Which of the following is the greatest possible common divisor of two [#permalink]
I typically struggle with questions like these because it seems difficult to start. BUT, let's give it a go.

Two distinct positive integers, lets call them x and y, are less than 124

My first instinct: Prime factorize 124. We get: 124 = 2 * 2 * 31

OK. next step, we need to find the GCF/HCF, so instinctively im looking at:

62. 62 is 31 * 2; say x = 62 and y = 62* 2 = 124. Well this won't work because y has to be less than 124.

That means the figure has to be 61. x = 61; 61/61 = 1 integer value; then y = 61 * 2 = 122; 122/61 = 2 Integer Value

The formula i have in my head is, x/hcf AND y/hcf should be an integer.

Therefore E
Re: Which of the following is the greatest possible common divisor of two [#permalink]
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