What is the greatest possible common divisor of two differen

The largest prime whose multiple of 2 is less than 144.

71 is prime.

71 * 2 < 144

there's your answer.

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?

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!

Thanks Vinay!

Yes, if the two numbers can be the same, then the numbers themselves will be the GCD and hence 143 will be the answer.

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.

Similar question to practice: what-is-the-smallest-possible-common-multiple-of-two-integer-130418.html

We need to find What is the greatest possible common divisor(GCD) of two different positive integers which are less than 144

Let's solve it using two methods

Method 1: Logic

Now the numbers are less than 144
=> We can take the numbers as 142 and 143, but then their GCD = 1 as they will have only 1 as the common factor.

If we think about this then we will realize that we can take two numbers such that one number is a big number (lets say x) *1 and other is a big number (lets say x) * 2
=> \(\frac{144}{2}\) = 72 can be taken as x and the numbers can be taken as 72*1 = 72 and 72*2 = 144

But, the numbers have to be less than 144
=> We can try with \(\frac{142}{2}\) = 71 and 71*2 = 142

So, Answer will be D

Method 2: Eliminate Option Choices

A. 143 For GCD of two different numbers to be 143 both have to be multiple of 143
=> We can take the numbers as 143*1 and 143*2 = 286, but numbers CANNOT be ≥ 144 => NOT POSSIBLE

B. 142 For GCD of two different numbers to be 142 both have to be multiple of 142
=> We can take the numbers as 142*1 and 142*2 = 284, but numbers CANNOT be ≥ 144 => NOT POSSIBLE

C. 72 For GCD of two different numbers to be 72 both have to be multiple of 72
=> We can take the numbers as 72*1 and 72*2 = 144, but numbers CANNOT be ≥ 144 => NOT POSSIBLE

D. 71 For GCD of two different numbers to be 71 both have to be multiple of 71
=> We can take the numbers as 71*1 and 71*2 = 142, which are both < 144 => POSSIBLE

So, Answer will be D
Hope it helps!

To learn more about LCM and GCD watch the following videos

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