Let m,n,z belong to N such that m and n divide z and gcd(m,n) = 1. Prove that m.n divides z.
To prove that m.n divides z, we need to show that z can be expressed as a product of m, n, and another natural number.
Given that m and n divide z, we can write:
z = a.m (for some natural number a)
z = b.n (for some natural number b)
To prove that m.n divides z, we need to find a natural number c such that z = c.(m.n).
Since gcd(m, n) = 1, m and n are coprime, which implies that there exist integers s and t such that:
1 = s.m + t.n
Multiplying both sides by z, we get:
z = s.m.z + t.n.z
Substituting the expressions for z in terms of m and n obtained earlier, we have:
z = s.(a.m).z + t.(b.n).z
Rearranging the equation, we get:
z = (s.a.z.m) + (t.b.z.n)
Now, let's define c = (s.a.z) = (t.b.z)
Therefore, we have:
z = c.(m.n)
Hence, we have shown that z can be expressed as a product of m, n, and the natural number c. This implies that m.n divides z, which completes the proof.