Find the Pumping Length for Language L of (2+3k) a's or (10+12k) b's

Question

The following question on the theory of computation is GATE 2019 CS question 24:

For $Σ = \{a, b\}$, let us consider the regular language: $$L = \{x \mid x = a^{2+3k} \text{ or } x = b^{10+12k}, k \geq 0\}$$ Which one of the following can be a pumping length (the constant guaranteed by the pumping lemma) for $L$?

(A) 3$\quad$(B) 5$\quad$(C) 9$\quad$(D) 24

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My Attempt

I tired to solve like this. I divide the minimum string possible into $x(y^i)z$ that is $a^2$ so getting i value $2$ but option not available.

Then I take second minimum $a^5$ i.e., taking $x=\epsilon$ and $y=a^5$ and $z=\epsilon$.

I am getting $i=5$. So is it correct answer??

Answer 1

I am getting i=5. So is it correct answer?

I am afraid I could not understand your argument and conclusion clearly.

Anyway, can you check if you are able to pump $a^2, a^5, b^{10}, b^{22}$ using the pumping length of your choice?

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Here are a few results stated in the form of exercises that you can prove and use to solve the problem in the question and beyond.

Assume $L$ is an arbitrary regular language.
Exercise 1. Show that there exists $p_0=p_0(L)\in \Bbb N^+$ such that $p\in\Bbb N$ is a pumping length for $L$ iff $p\ge p_0$.
We call $p_0$ the minimum pumping length of $L$.

Exercise 2. Let $L_{m,n}=\{a^m, a^{m+n}, a^{m+2n}, \cdots\}$ for $m,n\in\Bbb N$, $0<n$. Show that $p_0(L_{m,n})=m+1$.

Exercise 3. Let $L_1, L_2$ are two regular languages. Show that $p_0(L_1\cup L_2)\le\max(p_0(L_1), p_0(L_2))$

Exercise 4. Let $L_1, L_2$ are two regular languages over disjoint alphabets that do not contain the empty word. Show that $p_0(L_1\cup L_2)=\max(p_0(L_1), p_0(L_2))$

Exercise 5. Show that 11 is the minimum pumping length of $\{x\mid x=a^{2+3k} \text{ or } x=b^{10+12k},\ k\ge0\}$.