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. Understanding MOD Operation

The MOD operation calculates the remainder of the division of one number by another. In this case, we have: $$ z = x \MOD y $$ Where $ x = 23 $.

2. Setting up the equation

We need to find the values of $ y $ such that: $$ z = 2 $$ Thus, we need to set up the equation: $$ 23 \MOD y = 2 $$

3. Rearranging the equation

From the MOD operation, we know that for $ 23 \MOD y = 2 $, the equation can be rearranged to: $$ 23 = k \cdot y + 2 $$ for some integer $ k $. Simplifying gives: $$ 23 - 2 = k \cdot y $$ $$ 21 = k \cdot y $$

4. Finding candidates for y

To find potential values for $ y $, we note that $ y $ must be a divisor of $ 21 $. The divisors of $ 21 $ are $ 1, 3, 7, $ and $ 21 $. Next, we check which of these numbers 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. Final List of Valid y Values

The possible values of $ y $ that yield $ z = 2 $ are $ 3, 7,$ and $ 21 $.