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The only 3 consecutive odd numbers that are primes are 3,5,7

1. The problem statement, all variables and given/known data
Show that the only three consecutive numbers that are primes are 3,5,7.

2. Relevant equations

3. The attempt at a solution
let p, p+2, p+4 be three consecutive odd numbers
If p=0(mod3), p is divisible by 3
If p=1(mod 3), p+2 is divisible by 3
If p=2(mod3), p+4 is divisible by 3

This means at least one of p, p+2, p+4 is divisible by 3

Since we are looking for prime numbers 3 can be the only number that is divisible by 3. Therefore we only have 3 possible solutions:

-1,1,3
1,3,5
3,5,7

Since -1 and 1 are not primes the only possible solution is 3,5,7

-I no i have the solution here, its just i was helped with this and i dont quite understand why we bring in (mod3) is that just the way it is done or why do you include it??
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