# Answer to Question #42003 in Combinatorics | Number Theory for Janette

Question #42003

If p is a prime greater than 5, what are the possible values of p modulo 6?

Expert's answer

The p modulo 6 is the remainder of division of p by 6.

p = 6*(p/6) + (p%6),

where (p/6) is the integer quotient, and (p%6) is the modulo.

In general, the range of numbers for an integer modulo of n is 0 to n 1.

If p is a prime, this means p is divisible only by 1 and p. Since p is not divisible by 6, it would not leave a remainder of 0.

Also, a prime can not leave a remainder of 2, because then it would be divisible by 2:

p = 6*(p/6) + 2 = 2*(3*(p/6) + 1) and would not be a prime.

For the same reason, the remainder 3 and 4 are not possible.

Therefore, the possible modulos are 1 and 5.

Answer: 1, 5.

p = 6*(p/6) + (p%6),

where (p/6) is the integer quotient, and (p%6) is the modulo.

In general, the range of numbers for an integer modulo of n is 0 to n 1.

If p is a prime, this means p is divisible only by 1 and p. Since p is not divisible by 6, it would not leave a remainder of 0.

Also, a prime can not leave a remainder of 2, because then it would be divisible by 2:

p = 6*(p/6) + 2 = 2*(3*(p/6) + 1) and would not be a prime.

For the same reason, the remainder 3 and 4 are not possible.

Therefore, the possible modulos are 1 and 5.

Answer: 1, 5.

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