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What is the greatest possible common divisor of two different positive integers which are less than 144?

A. 143 B. 142 C. 72 D. 71 E. 12

Can someone explain why the answer is 71 if we assume that the integers are 143 and 142?

First of all, what is the greatest common divisor of 143 and 142? It is 1. You are looking for the common divisor. 142 and 143 will have no common divisor except 1.

Think: 2 and 3 have GCD (greatest common divisor) of 1 2 and 4 have GCD of 2. 3 and 4 have GCD (greatest common divisor) of 1 So if you were to select 2 numbers less than 5 with the greatest GCD, you need to select 2 and 4, not 3 and 4.

Now think: 143 = 11 * 13 The greatest possible divisor it will have with another number less than 144 will be either 11 or 13. Let's move on. 142 = 2*71 The greatest possible divisor it can have with another number less than 144 can be 71 (say, if the other selected integer is 71)

Do you think another number less than 144 could have a GCD of greater than 71? No because when you split a number into two factors, one of them will be at least 2. If it is greater than 2, the other factor will obviously be less than 71.

It's a very intuitive concept. Take some numbers to comprehend it fully. These posts will also be helpful:

Re: What is the greatest possible common divisor of two differen [#permalink]

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01 May 2013, 10:18

VeritasPrepKarishma wrote:

cv3t3l1na wrote:

What is the greatest possible common divisor of two different positive integers which are less than 144?

A. 143 B. 142 C. 72 D. 71 E. 12

Can someone explain why the answer is 71 if we assume that the integers are 143 and 142?

First of all, what is the greatest common divisor of 143 and 142? It is 1. You are looking for the common divisor. 142 and 143 will have no common divisor except 1.

Think: 2 and 3 have GCD (greatest common divisor) of 1 2 and 4 have GCD of 2. 3 and 4 have GCD (greatest common divisor) of 1 So if you were to select 2 numbers less than 5 with the greatest GCD, you need to select 2 and 4, not 3 and 4.

Now think: 143 = 11 * 13 The greatest possible divisor it will have with another number less than 144 will be either 11 or 13. Let's move on. 142 = 2*71 The greatest possible divisor it can have with another number less than 144 can be 71 (say, if the other selected integer is 71)

Do you think another number less than 144 could have a GCD of greater than 71? No because when you split a number into two factors, one of them will be at least 2. If it is greater than 2, the other factor will obviously be less than 71.

It's a very intuitive concept. Take some numbers to comprehend it fully. These posts will also be helpful:

Just to confirm, the only reason 143 is not the answer is because of the fact that the question mentioned 'two different positive numbers' right?
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Re: What is the greatest possible common divisor of two differen [#permalink]

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01 May 2013, 12:12

Quote:

Just to confirm, the only reason 143 is not the answer is because of the fact that the question mentioned 'two different positive numbers' right?

Let the two positive integers be a,b where both a,b<144. Now, let the required GCD be k. Thus, a = kM and b = kN, where M,N are positive integers and are not equal.

If k = 143, then the only way a<144 is if M = 1.Similarly, even for b, N=1. But as M is not equal to N, this is an invalid option.

The same for k=142 and 72.However, for k = 71, we can have M=1,N=2 OR M=2,N=1.

D.

If they wouldn't have mentioned that fact, we could have chosen the same value for M=N=1.
_________________

Re: What is the greatest possible common divisor of two differen [#permalink]

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02 May 2013, 08:12

vinaymimani wrote:

Quote:

Just to confirm, the only reason 143 is not the answer is because of the fact that the question mentioned 'two different positive numbers' right?

Let the two positive integers be a,b where both a,b<144. Now, let the required GCD be k. Thus, a = kM and b = kN, where M,N are positive integers and are not equal.

If k = 143, then the only way a<144 is if M = 1.Similarly, even for b, N=1. But as M is not equal to N, this is an invalid option.

The same for k=142 and 72.However, for k = 71, we can have M=1,N=2 OR M=2,N=1.

D.

If they wouldn't have mentioned that fact, we could have chosen the same value for M=N=1.

Got it! as you said...if the numbers could have been same, we could have used 143 as both the integers and the GCD wud have been 143!

What is the greatest possible common divisor of two different positive integers which are less than 144?

A. 143 B. 142 C. 72 D. 71 E. 12

Can someone explain why the answer is 71 if we assume that the integers are 143 and 142?

First of all, what is the greatest common divisor of 143 and 142? It is 1. You are looking for the common divisor. 142 and 143 will have no common divisor except 1.

Think: 2 and 3 have GCD (greatest common divisor) of 1 2 and 4 have GCD of 2. 3 and 4 have GCD (greatest common divisor) of 1 So if you were to select 2 numbers less than 5 with the greatest GCD, you need to select 2 and 4, not 3 and 4.

Now think: 143 = 11 * 13 The greatest possible divisor it will have with another number less than 144 will be either 11 or 13. Let's move on. 142 = 2*71 The greatest possible divisor it can have with another number less than 144 can be 71 (say, if the other selected integer is 71)

Do you think another number less than 144 could have a GCD of greater than 71? No because when you split a number into two factors, one of them will be at least 2. If it is greater than 2, the other factor will obviously be less than 71.

It's a very intuitive concept. Take some numbers to comprehend it fully. These posts will also be helpful:

Re: What is the greatest possible common divisor of two differen [#permalink]

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03 May 2013, 13:28

I am not entirely sharp because of a day of studying, but isn't it just that 71 is the GCD of x and y because x and y would then be respectably 71 and 142? Nothing more nothing less.

What is the greatest possible common divisor of two different positive integers which are less than 144?

A. 143 B. 142 C. 72 D. 71 E. 12

The key word here is different

If the two numbers were allowed to be the same, then we could use 143 and 143, in which case, the GCD = 143

However, since the two numbers must be DIFFERENT, then we might first try to do something with 143 and some other number. Since 143 = (11)(13), we can see that, in order to MAXIMIZE the GCD of the two numbers, the other number must be 13 So, 143 and 13 have a GCD of 13

Now let's try 142 and some other number. Since 142 = (2)(71), we can see that, in order to MAXIMIZE the GCD of the two numbers, the other number must be 71 So, 142 and 71 have a GCD of 71

Following this logic, we can see that 142 and 71 will MAXIMIZE the GCD of the 2 numbers.

So, the correct answer is D

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Re: What is the greatest possible common divisor of two differen
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