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If $frac{a}{b}$ is irreducible, then the quotient of the product of any $2$ factors of $a$ and any $2$ factors of $b$ are irreducible.

Mathematics Asked by Simplex1 on October 15, 2020

$a,bin mathbb{Z}$

Factors of $a$: $a_1,a_2,…,a_n$.

Factors of $b$: $b_1,b_2,…,b_m$

Prove that if $frac{a}{b}$ is irreducible, then
$frac{a_ia_j}{b_kb_l}$ is irreducible for all $i,j,k,l$.

I proved the case where $i=j, k=l$ using the fact that the square root of any non-perfect square is irrational.

I can’t prove the more general case though.

EDIT: not just prime factors, just all the integer factors. E.g. 12: 1,2,3,4,6,12

3 Answers

Alternative approach : proof by contradiction.

Suppose that $frac{a_i a_j}{b_k b_l}$ is not irreducible.
Then $exists ;$ prime $;p ;ni p|(a_i times a_j)$ and $p|(b_k times b_l) Rightarrow$
$p|a$ and $p|b Rightarrow frac{a}{b}$ is not irreducible.

Correct answer by user2661923 on October 15, 2020

If $frac{a_ia_j}{b_kb_l}$ were reducible then there would exist a prime number $pinmathbb{N}$ such that

$a_ia_j=pcdotalpha;$,

$b_kb_l=pcdotbeta;$.

So the prime number $p$ would be a factor of $a_i$ or $a_j$ and it would also be a factor of $b_k$ or $b_l$.

Consequently $p$ would be a factor of $a$ and a factor of $b$ too.

For this reason the fraction $frac{a}{b}$ would not be irreducible and it is a contradiction.

So it is impossible that $frac{a_ia_j}{b_kb_l}$ is reducible and it means that $frac{a_ia_j}{b_kb_l}$ is irreducible.

Answered by Angelo on October 15, 2020

If a prime number $p$ divides both $a_i a_j$ and $b_k b_l$, then it divides $a_i$ or $a_j$ hence $a$, and it also divides $b_k$ or $b_l$ hence $b$. This is impossible hence there is no such prime number and $frac{a_i a_j}{b_k b_l}$ is irreductible.

Answered by Gribouillis on October 15, 2020

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