<s>[INST] Show that $5n+3$ and $7n+4$ are relatively prime for all $n$. Show that $5n+3$ and $7n+4$ are relatively prime for all $n$.
 [/INST] Bezout's Lemma states that for if and only if $a$ and $b$ are comprime numbers then the following equation has integer solutions:
$$ax + by = 1$$
Now let $a=5n+3$ and $b=7n+4$. Now we get:
$$(5n+3)x + (7n+4)y = 1$$
Now apply the extended Euclidean Algorithm:
$$(7n+4) = (5n+3) + (2n+1)$$
$$(5n+3) = 2\times(2n+1) + (n+1)$$
$$(2n+1) = (n+1) + n$$
$$(n+1) = n + 1$$
We now just go back:
$$1 = (n+1) - n$$
$$1 = (n+1) - ((2n+1) - (n+1))$$
$$1 = 2(n+1) - (2n+1)$$
$$1 = 2((5n+3) - 2(2n+1)) - (2n+1)$$
$$1 = 2(5n+3) - 5(2n+1)$$
$$1 = 2(5n+3) - 5((7n+4) - (5n+3)$$
$$1 = 7(5n+3) - 5(7n+4)$$
We just obtained one solution $(x,y) = (7,5)$, but it's enough to prove that $7n+4$ and $5n+3$ are comprime numbers.
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