Concept doubt from number property

In Quant, you can establish innumerable inferences from the theory of any topic. The point is that you should be comfortable with the theory. If I give you a statement, you should be able to say whether it is true or false based on your conceptual understanding. There is no point memorizing these facts. Just try to understand why the book says it is so. Next time, if you come across a situation dealing with GCF, you don't need to 'recall' these; you will know that these are true.


1) Consecutive multiples of 'n' have a G.C.F of 'n'

What are consecutive multiples?
e.g. 4n, 5n or 18n, 19n etc are pairs of consecutive multiples of n. What will be the greatest common factor of 18n and 19n? We know that n is their common factor. Is there any common factor between 18 and 19 (except 1)? No. So GCF will be n only. Take any two consecutive numbers. They will have no common factor except 1. Hence, if we have two consecutive factors of n, their GCF will always be n.


2) The G.C.F of two numbers cannot be larger than difference between two numbers.
This is true in case the numbers are distinct. GCF is a factor of both the numbers. Say GCF of two distinct numbers is x. This means the two numbers are mx and nx where n and m have no common factor. What can be the smallest difference between m and n? m and n can be consecutive numbers. In this case, the difference between nx and mx will be x which is equal to the GCF. If m and n are not consecutive, the difference between mx and nx will be much larger than x. The difference between mx and nx cannot be less than x.
Say, GCF of two numbers is 6. The numbers can be 6 and 12 or 6 and 36 etc but they cannot be 6 and 8 since both numbers must have 6 as a factor.