Question

How can we show that if (a,b) =(b,c) = (a,c) =1and a/m, b/m, c/m then (abc)/m

Answer 1

Those are commos. This is a problem from number Theory and I think we can solve it using FTA. but I am confused.

Answer 2

I dont really understand the second part of the statement. Are you asking:

If (a,b)=(a,c)=(b,c)=1 and m is an integer divisible by a, b, and c, then (abc) divides m?

Answer 3

If that is the case, then you proceed as follows:

Since a,b, and c divide m, it follows that m is a common multiple of a,b, and c. Hence the least common multiple, which i will denote by lcm(a,b,c), divides m. Note that:\[\mbox{lcm}(a,b,c)=\mbox{lcm}(a,\mbox{lcm}(b,c)).\]Since (b,c)=1, and for any integers x and y:\[\gcd(x,y)\cdot \mbox{lcm}(x,y)=xy\]we deduce that:\[\mbox{lcm}(b,c)=bc\]Hence:\[\mbox{lcm}(a,b,c)=\mbox{lcm}(a,bc).\]Since (a,b)=(a,c)=1, (a,bc) must also equal 1. Therefore:\[\mbox{lcm}(a,bc)=abc.\] Hence abc divides m.