Could somebody help me understand this question?

"Show that if $a_1, a_2, ..., a_n$ are pairwise relatively prime integers, then $[a_1, a_2, ..., a_n]=a_1a_2...a_n$."

Question

Could somebody help me understand this question?

"Show that if $a_1, a_2, ..., a_n$ are pairwise relatively prime integers, then $[a_1, a_2, ..., a_n]=a_1a_2...a_n$."

jamesj · Answer

what's the definition of [a_1, ..., a_n] ?

anonymous · Answer

That is the least common multiple of those numbers.

jamesj · Answer

right.  I'm sure there's a couple of ways to do this.  But I would use prime factorization.  Suppose
$$ x = p_1^{a_1}p_2^{a_2}...p_n^{a_n}, \ \  y =  p_1^{b_1}p_2^{b_2}...p_n^{b_n} $$
where the $ p_i $ are distinct prime numbers and the
$$ a_1, a_2, ... a_n, b_1, ..., b_n $$
are non-negative integers. Then the l.c.m. of x and y
$$ [x,y] = \Pi_i \  p_i^{\max(a_i,b_i)} $$
For example, $ 12 = 2^23^1 $ and $ 50 = 2^15^2 $, then
$$ [12,5] = 2^23^15^2 = 300 $$

Now, use that definition for your problem and it should be straightforward.

anonymous · Answer

Perfect. I'll try that. Thank you very much.