greatest possible common divisor?

Question

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


i dont want solution first ...but i want to undersstand this question first.

it says "two different positive integers which are less than 144"

well, so can i assume 143 and 142 these two numbers in consideration ?

143 is prime ...so these two numbers can have 1 as common divisor .

we need the highest common divisor as such.


ok..lets move bit further ...say now we take 142,140 two numbers (excluding prime nos as they will contribute only 1)

so they can have 2 as the highest common divisor.



is this the process i should look for ? is this the questions wants me to seach for ?


Have i understood the question correctly ?

Answer 1

You have understood the question correctly.
I will give you a hint.
Take the largest even number less than 144
and take half of it. What do you get?
BTW 143 is not prime. It equals 11*13.

Answer 2

However we are looking for the largest common divisor and your idea of going for the division by two is more promissing indeed. we would need to different smallest divisors though instead of two only. If we go with 2 and 3 as divisors what is the largest factor we can mulitply by them and still be under 144
144/3=48 and 48 is too big because you say your number must be less than 144
but 47*3 gives 141
and 47*2 =94
Any factor larger than 47 would fail on the multplication by three.
What if we try 1 and 2 as the smallest factors.
144/2=72 is not less than 144 so
142/2=71
and 71/1=71
so using 71 and 142 gives a common factor of 71.
(but I suspect that dividing by one would be counted as a cheat)

Answer 3

Assuming that a "greatest common divisor" is the same thing as a "greatest common factor" (which I would assume), you are definitely on the right track. The terminology is a bit inconsistent since we speak of "least common divisors" (LCD) but "greatest common factors" (GCF). Looked at from the perspective of a GCF, you should come up with a solution quickly.