Prove that for all positive integers “a” and “b”, GCD(a,b) = LCM(a,b) iff a = b. Can someone explain me how to prove this? I'm confused. This is for Discrete Math.
To prove that GCD(a, b) = LCM(a, b) if and only if a = b, we need to show two separate implications:
1. If GCD(a, b) = LCM(a, b), then a = b.
2. If a = b, then GCD(a, b) = LCM(a, b).
Let's start with the first implication:
1. If GCD(a, b) = LCM(a, b), then a = b.
To prove this, we can use the properties of greatest common divisor (GCD) and least common multiple (LCM). In particular, we will use the property that GCD(a, b) * LCM(a, b) = a * b for any positive integers a and b.
Suppose GCD(a, b) = LCM(a, b). Then we have:
GCD(a, b) * LCM(a, b) = a * b (property of GCD and LCM)
Substitute a = b into the equation above:
GCD(a, a) * LCM(a, a) = a * a
GCD(a, a) * LCM(a, a) = a^2
a * a = a^2 (since GCD(a, a) = a and LCM(a, a) = a)
a^2 = a^2
This shows that if GCD(a, b) = LCM(a, b), then a = b.
Now, let's move on to the second implication:
2. If a = b, then GCD(a, b) = LCM(a, b).
To prove this, we can use the property that for any positive integer k, GCD(k, k) = LCM(k, k) = k.
Suppose a = b. Then we have:
GCD(a, b) = GCD(a, a) = a
LCM(a, b) = LCM(a, a) = a
This shows that if a = b, then GCD(a, b) = LCM(a, b).
Since we've proven both implications, we can conclude that GCD(a, b) = LCM(a, b) if and only if a = b.
well, you can start off by noting that
LCM(a,b) = ab / GCD(a,b)
see where that takes you.
Also, see what happens if a=b.