Author: Arun Sharma
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The remainder when a prime number p ≥ 5 is divided by 6 is 1 or 5. However, if a number on being divided by 6 gives a remainder of 1 or 5 the number need not be prime. Thus, this can be referred to as a necessary but not sufficient condition.
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The remainder of the division of the square of a prime number p ≥ 5 divided by 24 is 1.
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For prime numbers p > 3, p2 – 1 is divisible by 24.
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If a and b are any two odd primes then a2 – b2 is composite. Also, a2 + b2 is composite.
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The remainder of the division of the square of a prime number p ≥ 5 divided by 12 is
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Extending this logic, we can say that if we are not able to find a factor of a number upto the value of its square root, we will not be able to find any factor above the square root and the number under consideration will be a prime number.