Question

if a, b, c are any three integers such that (a,c)=1 and (b,c)=1, then show that (ab,c)=1.

Answer 1

Answer on question 33904 – Math – Number Theory

if a, b, c are any three integers such that (a,c)=1 and (b,c)=1, then show that (ab,c)=1.

Solution

Let the prime factorizations of a, b and c are

$$a = p _ {1} ^ {a _ {1}} \dots p _ {k} ^ {a _ {k}}, \qquad b = q _ {1} ^ {b _ {1}} \dots q _ {n} ^ {b _ {n}}, \qquad c = r _ {1} ^ {c _ {1}} \dots r _ {m} ^ {a _ {m}}$$

From the condition (a,c)=1 we get that for any $i = 1 \ldots k, j = 1 \ldots m$, $p_i \neq r_j$.

From the condition (b,c)=1 we get that for any $i = 1 \ldots m, j = 1 \ldots n$, $r_i \neq q_j$.

Then the prime factorization of ab is

$$a b = p _ {1} ^ {a _ {1}} \dots p _ {k} ^ {a _ {k}} q _ {1} ^ {b _ {1}} \dots q _ {n} ^ {b _ {n}}$$

As any of this multipliers doesn't equal to any of the multipliers of $c$ then (ab,c)=1.

QED.

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