How many two-digit prime numbers have a units digit of 7?
The tens digit cannot be 2, 5, or 8 because otherwise, the sum of the digits, and thus the number itself, is divisible by 3. The remaining possibilities are 17, 37, 47, 67, 77, and 97. Only $77=7\cdot11$ is composite, so there are $\boxed{5}$ two-digit prime numbers that have a units digit of 7.
5
Level 3
Prealgebra
Problem: How many two-digit prime numbers have a units digit of 7?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine how many two-digit prime numbers have a units digit of 7, we first identify the range of two-digit numbers, which is from 10 to 99. The two-digit numbers that end with a 7 are: 17, 27, 37, 47, 57, 67, 77, 87, and 97. 

Next, we need to check which of these numbers are prime. A prime number is defined as a number greater than 1 that has no positive divisors other than 1 and itself. 

We evaluate each candidate:
- 17: Divisors are 1 and 17. It is prime.
- 27: Divisors include 1, 3, 9, and 27. It is not prime.
- 37: Divisors are 1 and 37. It is prime.
- 47: Divisors are 1 and 47. It is prime.
- 57: Divisors include 1, 3, 19, and 57. It is not prime.
- 67: Divisors are 1 and 67. It is prime.
- 77: Divisors include 1, 7, 11, and 77. It is not prime.
- 87: Divisors include 1, 3, 29, and 87. It is not prime.
- 97: Divisors are 1 and 97. It is prime.

After evaluating all candidates, the prime numbers identified are 17, 37, 47, 67, and 97. 

Counting these, we find there are 5 two-digit prime numbers that have a units digit of 7. 

The final answer is 5.<|END_OF_LATENT|> The final answer is $5$.