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- Homework Statement
- Prove the assertion below:

Each integer of the form 3n+2 has a prime factor of this form.

- Relevant Equations
- None.

Proof: Suppose that all primes except for 3 must have

remainder of 1 or 2 when divided by 3.

Then we have the form 3p+1 or 3p+2.

Note that the product of integers of the form 3p+1

also have the form 3p+1.

Let m be an integer whose prime divisors have the form 3p+1,

now we know that m also has the form 3p+1.

Since the given integer has the form 3n+2,

it follows that not all of the prime divisors have the form 3p+1.

Thus, one of them will have the form 3p+2.

Therefore, each integer of the form 3n+2 has a prime factor of this form.

Above is my proof for this assertion. Can anyone please review/verify this and see if it's correct?

remainder of 1 or 2 when divided by 3.

Then we have the form 3p+1 or 3p+2.

Note that the product of integers of the form 3p+1

also have the form 3p+1.

Let m be an integer whose prime divisors have the form 3p+1,

now we know that m also has the form 3p+1.

Since the given integer has the form 3n+2,

it follows that not all of the prime divisors have the form 3p+1.

Thus, one of them will have the form 3p+2.

Therefore, each integer of the form 3n+2 has a prime factor of this form.

Above is my proof for this assertion. Can anyone please review/verify this and see if it's correct?