Disprove the following statements for positive integers $a$, $b$, and $c$:
(a) $[p, p] = p^2$ where $p$ is prime.
(b) $[a, b] = a × b$
(c) If $[a, b] = n$ and $[b, c] = m$ then $[a, c] = m × n$.
(d) $[a + b, c] = [a, c] + [b, c]$
(e) $[ab, ac] = a^2 [b, c]$
(f) $\gcd (a, b, c) × [a, b, c] = a × b × c$
We disprove all the statements with counter-examples.
(a) Choosing $p=2$
$ [2,2] = 2 \ne 4 = 2^2 $
(b) Choosing $a=2, b=4$
$ [2,4] = 4 \ne 8 = 2 \times 4 $
(c) Choosing $a=1, b=2, c=1$
$n = [1,2] = 2$
$m = [2,1] = 2$
$[1,1]=1 \ne 4 = 2 \times 2$
(d) Choosing $a=1, b=2, c=3$
$[1+2,3] = 3 \ne 9 = 3 + 6= [1,3] + [2,3]$
(e) Choosing $a=2, b=1, c=1 $
$ [2, 2] = 2 \ne 4 = 2^2 \times [1,1] $
(f) Choosing $a=2, b=2, c=2$
$\gcd(2, 2, 2) \times [2,2,2] = 2 \times 2 = 4 \ne 8 = 2 \times 2 \times 2$