Which of the following initial values of the variable y would result in the variable z being set to 2 after the code segment is executed?

Steps to Solve

1. Understand the Modulus Operation

The modulus operation $z = x \mod y$ gives the remainder when $x$ is divided by $y$. We need to find values of $y$ such that when $23$ is divided by $y$, the remainder is $2$.

2. Set Up the Equation

From the problem, we can set up the equation: $$ 23 \mod y = 2 $$

This means that the expression $23 - 2$ needs to be divisible by $y$.

3. Calculate the Divisor

Rearranging the equation gives: $$ 21 = 23 - 2 $$

So, we need to find values of $y$ that divide $21$. The possible values of $y$ are the divisors of $21$.

4. Identify Divisors of 21

The divisors of $21$ are $1$, $3$, $7$, and $21$. We'll also check which of these, when substituted into $x \mod y$, results in $z = 2$:

• For $y = 1$: $23 \mod 1 = 0$ (not valid)
• For $y = 3$: $23 \mod 3 = 2$ (valid)
• For $y = 7$: $23 \mod 7 = 2$ (valid)
• For $y = 21$: $23 \mod 21 = 2$ (valid)

5. Select from Given Options

The problem gives us the following options for $y$:

• A. 1
• B. 2
• C. 3
• D. 4

From our identified values, only $y = 3$ meets the requirement from the options.