Math 300

Fraction Unit

Dr. Wilson

 

Fact: If a whole number, d, goes into a product, ab, and if the greatest common factor of a and d is 1, then d goes into b.

 

Proof: Suppose that d goes into ab. Say ab = md. If the greatest common factor of a and d is 1, then we can express 1 as a difference of multiples of a and d. Say that 1 = ra + sd where r and s are integers. Naturally, one of r and s will be negative and the other will be positive, but if we assume that they are integers, we will not have to worry about which one is positive and which one is negative. Then

 

b = 1.b

 

= (ra + sd)b

 

= rab + sdb

 

But we just said that ab = md. If we substitute that in we get

 

b = rmd + sbd

 

= (rm + sb)d

 

and we see that b is a multiple of d.

 

Definition: If the greatest common factors of two numbers is 1, then we say that the two numbers are relatively prime.

 

Note that the result need not be true if we drop the condition that d be relatively prime to one of the factors. For instance 6 goes into 24 which is 3x8, but 6 does not go into either 3 or 8. The reason that the last fact does not apply is because 6 is not relatively prime to either 6 or 8.