Question

If$p>5$ is prime and p is divided by 10, show that the remainder is 1, 3, 7, or 9.

Short answer

It is proved that if $p>5$ is prime and p is divided by 10, then remainders are 1, 3, 7, or 9.

Step-by-step solution

Write the possible reminder when a number is divided by 10

When any number is divided by 10, the possible remainders are$0,1,2,3,....,9$.

Show that If p>5 is prime and p is divided by 10, then remainders are1, 3, 7, or 9.

If the number p is divided by 10 and the possible remainders are 0, 2, 4, 6,and 8, this implies that p is even.

$p>2$and p is divisible by 2 in addition to 1 and itself and cannot be prime.

If the remainder is 5, then as $p>5$,p is divisible by 5 in addition to 1 and itself and cannot be prime.

This implies that the possible remainders are 1, 3, 7, or 9.

Hence, it is proved that if$p>5$ is prime and p is divided by 10, then remainders are1, 3, 7, or 9.