For prime numbers greater than 5, how many different digits are possible in the units place?

Introduction / Context:
This conceptual question asks about the units digits that prime numbers greater than 5 can have. It aims to test understanding of divisibility rules and the basic structure of prime numbers in the decimal system.

Given Data / Assumptions:

- We consider only prime numbers greater than 5.

- We are interested in the digit in the units place (the last digit).

- We must count how many different digits can appear as this last digit.

Concept / Approach:
If a number ends with an even digit (0, 2, 4, 6, 8), it is divisible by 2 and thus cannot be prime unless it is 2 itself. If a number ends with 5 or 0, it is divisible by 5 and cannot be prime unless it is 5 itself. Since we are considering primes greater than 5, their units digits must avoid these divisibility traps.

Step-by-Step Solution:
Step 1: List all possible units digits from 0 to 9. These digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Step 2: Exclude digits that force divisibility by 2 for numbers greater than 2. Digits 0, 2, 4, 6, and 8 at the end of a number make it even. Any such number greater than 2 is composite, not prime. Step 3: Exclude digits that force divisibility by 5 for numbers greater than 5. Digits 0 and 5 at the end of a number make it divisible by 5. Any such number greater than 5 is composite, not prime. Step 4: For prime numbers greater than 5, the only possible last digits are 1, 3, 7, and 9. So there are 4 possible units digits.

Verification / Alternative check:
Look at some examples: Primes greater than 5 include 7, 11, 13, 17, 19, 23, 29, 31, 37, and so on. Their units digits are 7, 1, 3, 7, 9, 3, 9, 1, 7, and so on, always from the set {1, 3, 7, 9}. This confirms that no other units digit occurs for such primes.

Why Other Options Are Wrong:
- 3: Too small; we have identified four possible digits (1, 3, 7, 9).
- 5: Too large; there are not five distinct digits that can appear at the end of primes greater than 5.
- 6: Also too large; it includes digits that would make the number composite.

Common Pitfalls:
Some learners mistakenly include 2 and 5 as possible units digits for primes in general without noting the restriction “greater than 5”. Others forget that ending in 0 implies divisibility by 10. Understanding divisibility rules and the exceptions (2 and 5 themselves) is the key to this question.

Final Answer:
The number of possible digits that can appear in the units place of a prime number greater than 5 is 4.