Consider a string consisting of $a^{\text{'}}s,b^{\text{'}}s,$ and $c^{\text{'}}$ s$,$ where the number of $b^{\text{'}}$ s is three times the number of $a^{\text{'}}$ s and the number of $c^{\text{'}}$ s is

five times the number of $a^{\text{'}}$ s$.$ Prove that the length of the string is divisible by $3.$

Proof: Suppose $s$ is a string of length $n$ that consists of $a^{\text{'}}s,b^{\text{'}}s,$ and $c^{\text{'}}s,$ where the number of $b^{\text{'}}s$ is

times the

number of $a^{\text{'}}s,$ and the number of $c\text{'}$s is

times the number of a$\text{'}$s$.$

Let $x,y$ and $z$ be the numbers of $a^{\text{'}}s,b\text{'}$s$,$ and $c^{\text{'}}s$ in $s,$ respectively.

The length of $s$ is the $---$ Select$---vv$ of the numbers of $a^{\text{'}}s,b\text{'}$s and $c^{\text{'}}s$ that are in $s.$ Hence,

Because $x$ is an integer, so is $(,)x,$ and thus $n$ equals $3*($an integer$).$

Hence, by definition of divisibility, $n$ is divisible by $3.$