Let $a=\prod_{i=1}^k p_i^{a_i}, b=\prod_{i=1}^k p_i^{b_i}$ with $a_i\ge0,b_i\ge0$. Show $(a,b)=1$ if and only if $a_ib_i=0$ for all $i,1\le i\le k,$

Asked May 05 '23 at 23:23

Active May 06 '23 at 02:10

Viewed 49 times

Let $a=\prod_{i=1}^k p_i^{a_i}$ and $b=\prod_{i=1}^k p_i^{b_i}$ with $a_i\ge0$ and $b_i\ge0$. Show that $(a,b)=1$ if and only if $a_ib_i=0$ for all $i,1\le i\le k, p_i primes$

I know I have to arrive at $a_ib_i=0$ , so $a_i=0$ or $b_i=0$.

If I start with the definition of greatest common divisor and with Bezout I was not able to reach the conclusion. please help.

edited May 06 '23 at 02:10

asked May 05 '23 at 23:23

Adriana Gonzalez

Please do not rely on pictures of text. Here's a MathJax tutorial :) – Shaun May 05 '23 at 23:28
1

If you know that $p_i$ are primes then you should say so in the question. Otherwise the "if" part does not hold true. – dxiv May 05 '23 at 23:54

Let $a=\prod_{i=1}^k p_i^{a_i}, b=\prod_{i=1}^k p_i^{b_i}$ with $a_i\ge0,b_i\ge0$. Show $(a,b)=1$ if and only if $a_ib_i=0$ for all $i,1\le i\le k,$

0 Answers0