7. 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?
Answer: C
Two 2-digit positive integers meet the criteria.
The 2-digit positive integers that have a remainder of 3 when divided by both 10 and 6 include 13 and 63. Thus, there are exactly two such integers.
A) None
This option is incorrect because there are indeed 2-digit integers that satisfy both conditions. Specifically, 13 and 63 meet the requirement of having a remainder of 3 when divided by both 10 and 6.
B) One
This option is also incorrect. While one might assume that only one number fits the criteria, there are actually two integers (13 and 63) that satisfy the conditions set by the question.
C) Two
This option is correct as there are two 2-digit integers, 13 and 63, that have a remainder of 3 when divided by both 10 and 6.
D) Three
This option is incorrect as there are only two integers that meet the specified conditions. No additional integers beyond 13 and 63 fulfill both remainder requirements.
E) Four
This option is incorrect because there are not four integers that satisfy the given conditions. The only relevant integers are 13 and 63.
Conclusion
The correct answer is two because the only two-digit integers that leave a remainder of 3 when divided by both 10 and 6 are 13 and 63. All other options fail as they either overestimate or underestimate the actual count of qualifying integers.