In the set N of natural numbers, define the binary operation * by m *

1.	In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?
Answer» The operation is clearly commutative since m * n = g.c.d (m, n) = g.c.d (n, m) = n * m ∀m, n ∈ N. It is also associative because for l, m, n ∈ N, we have l * (m * n) = g. c. d (l, g.c.d (m, n)) = g.c.d. (g. c. d (l, m), n) = (l * m) * n.

In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?

Answer»

The operation is clearly commutative since

m * n = g.c.d (m, n) = g.c.d (n, m) = n * m ∀m, n ∈ N.

It is also associative because for l, m, n ∈ N, we have

l * (m * n) = g. c. d (l, g.c.d (m, n))

= g.c.d. (g. c. d (l, m), n)

= (l * m) * n.