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

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Math Expert
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Which of the following is the greatest possible common divisor of two  [#permalink]

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12 Jul 2016, 03:37
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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.

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Which of the following is the greatest possible common divisor of two  [#permalink]

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12 Jul 2016, 04:48
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.

Given to find the greatest possible common divisor for two different positive number.

Let X be GCD and A and B are two different number.

We need to see the XA and XB has to be less than 124.

Now from options,

if we consider 62 * 2 = 124, which is equal to 124, we need to consider some number lesser than 124.

Consider 61 * 2 = 122 and 61 * 1 = 61 ( Both these numbers are lesser than 124 as requested ).

So GCD will be 61.

IMO E is correct option.

For clarity purpose let's check all the options.

A. 123 = ( 123 * 1 ) ( 123 * 0) can't be this. ( value = GCD * some number) . Apart from 0 and 1 , we'll get more result which is > 124 ( for ex 123 * 2 = 246 )
B. 122 = ( 122 * 1 ) ( 122 * 0 )
C. 63 = ( 63 * 1 ) ( 63 * 0 )

OA please...will correct if I missed anything.
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Re: Which of the following is the greatest possible common divisor of two  [#permalink]

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12 Jul 2016, 05:06
e.
61,122 because rest all are greater than 124 for their second multiple only.
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Re: Which of the following is the greatest possible common divisor of two  [#permalink]

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29 Apr 2017, 04:46
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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20 May 2017, 11:00
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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]

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05 Feb 2019, 10:01
chetan2u: i am unable to understand the reasoning behing it. Why it can't be 123?

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

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05 Feb 2019, 10:28
1
kaulmeankit08 wrote:
chetan2u: i am unable to understand the reasoning behing it. Why it can't be 123?

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]

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06 Feb 2019, 00:08
chetan2u wrote:
kaulmeankit08 wrote:
chetan2u: i am unable to understand the reasoning behing it. Why it can't be 123?

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]

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06 Feb 2019, 18:42
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.

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Re: Which of the following is the greatest possible common divisor of two   [#permalink] 06 Feb 2019, 18:42