SOLUTION: Could you please help me prove the following mathematical statement. Let p be a prime number and a and b positive integers, such that p/ab. then p/a or p/b.

Algebra ->  Divisibility and Prime Numbers  -> Lessons -> SOLUTION: Could you please help me prove the following mathematical statement. Let p be a prime number and a and b positive integers, such that p/ab. then p/a or p/b.       Log On


   

Question 870785: Could you please help me prove the following mathematical statement.
Let p be a prime number and a and b positive integers, such that p/ab. then p/a or p/b.


Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
If p | a, we are done.

Otherwise, p and a are relatively prime, so by Bezout's identity, there exist integers m and n such that pm + an = 1. Multiplying both sides by b, pmb + abn = b. Since pmb and abn are divisible by p (p | ab), then the sum pmb + abn is divisible by p, so b is divisible by p.