When x is divided by 2, remainder is 1 and y is divided by 8, remainder is 2. Which of...

1. Translate the problem requirements: When x is divided by 2, remainder is 1 means \(\mathrm{x = 2k + 1}\) for some integer k. When y is divided by 8, remainder is 2 means \(\mathrm{y = 8m + 2}\) for some integer m. We need to find which answer choice could be a value of \(\mathrm{2x + y}\).
2. Express variables in remainder form: Write x and y in terms of their quotients and remainders to create a general formula for \(\mathrm{2x + y}\).
3. Simplify the expression for \(\mathrm{2x + y}\): Substitute the remainder forms into \(\mathrm{2x + y}\) and identify the pattern of possible remainders.
4. Test answer choices against the pattern: Check which answer choices satisfy the remainder pattern we derived for \(\mathrm{2x + y}\).

Execution of Strategic Approach

1. Translate the problem requirements

When we say "x divided by 2 has remainder 1," x is odd: \(\mathrm{1 ÷ 2 = 0 \text{ remainder } 1}\), \(\mathrm{3 ÷ 2 = 1 \text{ remainder } 1}\), etc. So x = 1, 3, 5, ...
When we say "y divided by 8 has remainder 2," y = 2, 10, 18, ...
Mathematically:

• \(\mathrm{x = 2k + 1}\)
• \(\mathrm{y = 8m + 2}\)

2. Express variables in remainder form

Substitute into \(\mathrm{2x + y}\):
\(\mathrm{2(2k + 1) + (8m + 2)}\)

3. Simplify the expression

\(\mathrm{2x + y = 4k + 2 + 8m + 2 = 4k + 8m + 4 = 4(k + 2m + 1)}\)
So \(\mathrm{2x + y}\) is always a multiple of 4.

4. Test answer choices

1. 10: \(\mathrm{10 ÷ 4 = 2 \text{ remainder } 2}\) ✗
2. 11: \(\mathrm{11 ÷ 4 = 2 \text{ remainder } 3}\) ✗
3. 12: \(\mathrm{12 ÷ 4 = 3 \text{ remainder } 0}\) ✓
4. 13: \(\mathrm{13 ÷ 4 = 3 \text{ remainder } 1}\) ✗
5. 14: \(\mathrm{14 ÷ 4 = 3 \text{ remainder } 2}\) ✗

Only choice C (12) is a multiple of 4. Verify with x = 1, y = 10: \(\mathrm{2(1) + 10 = 12}\), and remainders check out. Answer: C) 12.