If 'a' is a prime number, then a² has exactly 3 factors.
Explanation
Let's break down why this is the case:
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Prime Number: A prime number is a whole number greater than 1 that has only two factors (divisors): 1 and itself. Examples include 2, 3, 5, 7, 11, etc.
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Factors of a²: When you square a prime number 'a', you get a². The factors of a² will always be:
- 1 (since 1 divides every number)
- a (the prime number itself)
- a² (the square of the prime number)
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Example: Let's take the prime number 3. Then a² would be 3² = 9. The factors of 9 are 1, 3, and 9.
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General Case: Since 'a' is prime, it has no other factors besides 1 and itself. Therefore, a² can only be divided by 1, 'a', and 'a²'.
Table Example
| Prime Number (a) | a² | Factors of a² | Number of Factors |
|---|---|---|---|
| 2 | 4 | 1, 2, 4 | 3 |
| 3 | 9 | 1, 3, 9 | 3 |
| 5 | 25 | 1, 5, 25 | 3 |
| 7 | 49 | 1, 7, 49 | 3 |
| 11 | 121 | 1, 11, 121 | 3 |
Conclusion
In conclusion, if 'a' is a prime number, then 'a²' will always have exactly three factors: 1, 'a', and 'a²'.
