I'm getting back into basic proofs after a long hiatus, and I know something has to be wrong with the following logic but I'm not sure what.
Elements of arithmetic progressions can be expressed as:
$a_n = a_0 + nd_a$
Where $a_n$ is the $n$th element of the progression and $d_a$ is the constnt difference or 'step.' Note that typically it's written with $(n - 1)d_b$ and indexed from 1 but as a programmer I find that silly. Now, Bezout's identity states any linear combination is equal to a multiple of the gcd of the two numbers being combined, so it follows that:
$a_0 + nd_a = gcd(a_0, d_a) * k$
Where $k$ is used to represent some satisfying multiple. We can rewrite this as an assertion about divisibility:
$gcd(a_0, d_a) \mid a_0 + nd_a$
Then since you can reverse any divisibility formula and swap $lcm$ for $gcd$:
$a_0 + nd_a \mid lcm(a_0, d_a)$
But I know this can't be true based on a simple counter example. Consider the progression that starts at 3 and increments by 4: $3, 7, 11, ...$
Our formula states that all numbers in the progression have to divide $lcm(3, 4) = 12$, which is not true for 7 or 11. There needs to be a growing coefficient for the lcm, but I'm not sure where it would have come in. An assumption in the proof for being able to swap lcm/gcd in divisibility formulas that I'm breaking?