Understanding Prime Numbers Greater Than 6
When considering prime numbers greater than 6, it’s essential to analyze how they behave when divided by 6.
Properties of Prime Numbers
- A prime number is defined as a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
- All prime numbers greater than 3 are odd, as even numbers greater than 2 can be divided by 2.
Dividing by 6: The Remainders
- Any integer can be expressed in terms of its remainder when divided by another integer. The remainders when dividing by 6 can be 0, 1, 2, 3, 4, or 5.
- The prime numbers greater than 6 must be examined under modulo 6:
- If a prime number is 6k: This number is not prime (it’s divisible by 6).
- If a prime number is 6k + 1: This number could be prime (e.g., 7, 13).
- If a prime number is 6k + 2: This is even and greater than 2, so it cannot be prime.
- If a prime number is 6k + 3: This number is divisible by 3, hence not prime.
- If a prime number is 6k + 4: This is also even and greater than 2, thus cannot be prime.
- If a prime number is 6k + 5: This number could be prime (e.g., 5, 11).
Conclusion
From the analysis above, the only possible remainders when a prime number greater than 6 is divided by 6 are:
- 1 (6k + 1)
- 5 (6k + 5)
Thus, the remainder when a prime number greater than 6 is divided by 6 is either 1 or 5.