Question 117931: Is it possible for the remainder to be 2 when a prime number that is greater than 2 is divided by 4? Explain why or why not?
Found 2 solutions by stanbon, Earlsdon:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Is it possible for the remainder to be 2 when a prime number that is greater than 2 is divided by 4? Explain why or why not?
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Is there a solution to x = 2(mod 4) if x is prime and greater than 2 ??
If so there is an odd prime where 4 divides (x-2) with no remainder.
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Argument:
If x is odd, x-2 is odd
But if x-2 is divisible by 4 it is a multiply of 2
Then x-2 would be even.
This contradiction means the assuption that there is
an odd prime x where x= 2(mod 4) is wrong.
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Cheers,
Stan H.
Answer by Earlsdon(6294) (Show Source):
You can put this solution on YOUR website! Not possible!
Why, because, by definition, a prime number is divisible only by 1 and itself.
If a number is divisible by 4, then it is not a prime number, because then the number would have three or more factors (1, 4 and the number itself and possibly others), whereas a prime number has only two factors.
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