Which of the following numbers are congruent to 2 (mod 3)
?

Question 11Select one:

a.
5,7,9,11,13


b.
5,8,11,14,17


c.
1,4,7,10,13


d.
1,3,5,7,9

To determine which of the given sets of numbers are congruent to \( 2 \ (\text{mod} \ 3) \), we need to check which numbers in each set give a remainder of 2 when divided by 3.

Let's evaluate each set:

**Option A: 5, 7, 9, 11, 13**
- \( 5 \div 3 = 1\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 7 \div 3 = 2\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 9 \div 3 = 3\) remainder \(0\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 11 \div 3 = 3\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 13 \div 3 = 4\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)

**Option B: 5, 8, 11, 14, 17**
- \( 5 \div 3 = 1\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 8 \div 3 = 2\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 11 \div 3 = 3\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 14 \div 3 = 4\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 17 \div 3 = 5\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)

**Option C: 1, 4, 7, 10, 13**
- \( 1 \div 3 = 0\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 4 \div 3 = 1\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 7 \div 3 = 2\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 10 \div 3 = 3\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 13 \div 3 = 4\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)

**Option D: 1, 3, 5, 7, 9**
- \( 1 \div 3 = 0\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 3 \div 3 = 1\) remainder \(0\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 5 \div 3 = 1\) remainder \(2\) → congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 7 \div 3 = 2\) remainder \(1\) → not congruent to \( 2 \ (\text{mod} \ 3) \)
- \( 9 \div 3 = 3\) remainder \(0\) → not congruent to \( 2 \ (\text{mod} \ 3) \)

**Correct Answer:**
The correct set is **Option B: 5, 8, 11, 14, 17**. All the numbers in this set are congruent to \( 2 \ (\text{mod} \ 3) \).

Would you like further details or have any questions?

Here are some related questions:
1. What does it mean for a number to be congruent modulo \( n \)?
2. How do you find the remainder of a division?
3. Which numbers are congruent to \( 1 \mod 4 \) from the set \( 6, 9, 12, 15, 18 \)?
4. How can you tell if two numbers are congruent modulo \( n \)?
5. What is the smallest positive number congruent to \( 2 \mod 5 \)?

**Tip:** When checking for congruence modulo \( n \), simply divide by \( n \) and check the remainder.

Explore how to identify numbers congruent to 2 modulo 3 using modular arithmetic. This problem involves evaluating sets of numbers and determining which ones give a remainder of 2 when divided by 3. Suitable for high school students.

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