Integers

[heshamelaziry]

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

143

142

72

71

12

[thephoenix]

IMO 71

[papgust]

Take a maximum integer which is less than 144. Avoid 143 as it is a prime number.
So, consider 142. 142 has 2 prime factors => 2 and 71.

For a number less than 144 to have a 71 as prime factor, it is the number 71 itself.
So, GCD of 71 and 142 = 71 (Option D)

[ssuarezo]

Hi:

I always get confused with the untold part. The stem question makes no restrictions about primes.
Question: why can't be 143 (143x1), even if prime, it would be the greatest number (as asked).

Thank you,
Silvia

[papgust]

When you have 143 and any other number (say 71), GCD of 143 and 71 will be 1. Because they have only a greatest common factor of 1. 143 has only 2 factors - 1, 143. 143 cannot be a factor for some other integer less than 144.

Remember, we are asked to find the greatest possible common divisor. So, we are ignoring 143 in this problem

[thephoenix]

Hi:

I always get confused with the untold part. The stem question makes no restrictions about primes.
Question: why can't be 143 (143x1), even if prime, it would be the greatest number (as asked).

Thank you,
Silvia

for 143 and 1 , 143 cant be the greatest common divisor as 143 doesnt divide 1 the GCD in this case will be 1

HTH

[Ian Stewart]

143 is not prime; it is equal to 11x13.

[ssuarezo]

Thank you Ian, Phoenix and Papgust.
True. They asked for the greatest common divisor, not the greatest number, and even if 143 is not prime ( didn't see it!), 11 and 13 are much lower than 71. I have to read carefully.
Thank you,
Silvia.