What is the greatest possible common divisor of two differen

Question

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?

KarishmaB · Accepted Answer

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.

BrentGMATPrepNow · Answer

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

RELATED VIDEO FROM OUR COURSE
 https://www.youtube.com/watch?v=1SbimzCkxQ0

AlyoshaKaramazov · Answer

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

71 is prime.

71 * 2 < 144

there's your answer.

arpanpatnaik · 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?

mau5 · Answer

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.

arpanpatnaik · Answer

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!

KarishmaB · Answer

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

Kwintessens · 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.

Bunuel · Answer

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

BrushMyQuant · Answer

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

https://www.youtube.com/watch?v=7jPV6JD0908

https://www.youtube.com/watch?v=vtSsAjcF-0Y

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