How many 2-digit positive integers have a remainder of 3 when divided by 10 and also have a remainder of 3 when divided by 6?

Rationale

Two-digit positive integers that meet the criteria are 13 and 33.

To find the two-digit integers that leave a remainder of 3 when divided by both 10 and 6, we can express these integers mathematically. The numbers that meet the first condition can be represented as \(10k + 3\) where \(k\) is an integer. To satisfy the second condition, these integers must also meet \(10k + 3 \equiv 3 \mod 6\), which simplifies to \(10k \equiv 0 \mod 6\) or \(4k \equiv 0 \mod 6\). This means \(k\) must be a multiple of 3, leading to the valid two-digit integers being 13 and 33.

A (None) YOUR ANSWER

This choice suggests that there are no two-digit integers fulfilling the given conditions. However, since we have identified at least two integers (13 and 33) that satisfy both conditions, this option is incorrect.

B (One) YOUR ANSWER

This option implies there is only one two-digit integer meeting the criteria. However, our analysis shows there are actually two integers, 13 and 33, that fulfill both conditions, making this choice incorrect.

C (Two) CORRECT

This is the correct choice, as we have explicitly identified two two-digit integers: 13 and 33. Both numbers leave a remainder of 3 when divided by 10 and also leave a remainder of 3 when divided by 6.

D (Three) YOUR ANSWER

This choice indicates that three two-digit integers meet the criteria. Given our findings, only two integers (13 and 33) satisfy the conditions, thus making this option incorrect.

E (Four) YOUR ANSWER

This option claims that there are four integers that meet the conditions. However, our analysis confirms only two valid integers, thereby rendering this choice incorrect.

Conclusion

In summary, the two-digit integers that leave a remainder of 3 when divided by both 10 and 6 are specifically 13 and 33. This demonstrates that while the conditions may seem limiting, there are indeed two valid solutions, confirming choice C as the only correct answer. All other options incorrectly represent the total count of integers that meet the specified criteria.

How many 2-digit positive integers have a remainder of 3 when divided by 10 and also have a remainder of 3 when divided by 6?

Two-digit positive integers that meet the criteria are 13 and 33.

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