This is a somewhat simple proof since it uses the definition of a greatest common divisor.

A greatest common divisor (GCD) is the largest integer which divides two other integers. Take for example the numbers 24 and 16. 8 is the GCD for 24 and 16 since 8 is the largest number that will divide both of them.

*So if the GCD=d then d divides a, and d divides b.*

The numbers 4 and 2 also divide 24 and 16, but are not the "Greatest". However, we do see that 4 and 2 also divide 8, which is the second part of the definition of a GCD.

*For every positive integer c, if c divides a and c divides b, then c divides d.*

It is this last part of the definition we will use for our proof. If c is an integer which divides a and b, then by definition c must also divide the GCD, which is d. And therefore c < d. ~~~~

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