This automata recognizes the following language:
\[
(a^5 + a^7)^*
\]
This accepts all strings whose lengths can be expressed as combinations of 5 and 7:
like \(a^5\), \(a^7\),\(a^{12}\),\(a^{14}\),etc.
\[
\text{i.e, it accepts all } a^n \text{ such that } n = 5x + 7y,\ x, y \geq 0
\]
so initially I was checking with options 18, 22, 23 and a few larger numbers, trying to find a pattern if answer was \(E\). While searching online for a better way to reason about it, I came across this beautiful result called the Frobenius Coin Problem, which saved a lot of effort. Proof for anyone interested.
It states:
For positive integers \(x, y\) that are relatively prime, every integer
\[
n \geq (x - 1)(y - 1)
\]
can be written as \(xa + yb\), for some non-negative integers \(a, b \geq 0\).
So applying it here:
\[
n \geq (5 - 1)(7 - 1) = 4 \cdot 6 = 24
\]
can all be expressed as \(5x + 7y\), and thus 23 is the largest value that cannot be expressed and that would also be the largest \(l\) that our FA cannot accept.
\[
\boxed{l = 23 \text{ is the length of the longest string that is NOT accepted by the above automata.}}
\]
