I'm taking a mathematics class where we have learned some introductory number theory - but I am having trouble with the whole 'proving this and that' component (most of it lol). Particularly with regards to exercises about common divisors and multiples. Consider the following example:
'Let a, b, d, be an element of the natural numbers. If d is a common divisor of a and b, then ab/d is a common multiple of a and b.' Prove this.
I started by supposing a, b, and d are natural numbers such that d is a common divisor of a and b. Then I get stuck - I do not know how to express d being a common divisor of a and b as a mathematical definition (what I mean by definition is that, for example, the definition of even is n = 2k for some integers n and k)
For d to be a common divisor of a and b, would it mean that:
a = dn+r for some integers n and r
b = dq+s for some integers q and s
by the quotient remainder theorem?
I'm just really lost and this is a last resort, I know you guys don't like 'homework' questions but if you can answer it generally that's fine.
Thanks in advance!