Question

Show that one and only one out of n, n+4, n+8, n+6 and n+12 is divisible by 5, where it is a positive integer.

Answer 1

Hint: A number is said to be divisible by 5 if we get remainder zero while we are dividing that number by 5. If we divide n by 5 we can write n = 5q + r, where q is the quotient and r is the remainder and r can be 0, 1, 2, 3, 4. Take every possible value for r as a separate case and check if n, n+4, n+8, n+6, n+12 is divisible by 5 or not.

Complete step-by-step answer:
We know that a number is said to be divisible by 5 if we get remainder zero while we are dividing that number by 5.

Suppose, we are dividing n by 5, where n is any positive integer. Let the quotient be q and the remainder be r. Then we can write,
n = 5q+r, where r can be 0, 1, 2, 3, 4.
Therefore, n = 5q+0 or n = 5q+1 or n = 5q+1 or n = 5q+2 or n = 5q+3 or n = 5q+4.
When,
 n = 5q, n is divisible by 5.
n+4 = 5q+4, is not divisible by 5 as the remainder is 4.
n+8 = 5q+8 = (5q+5)+3, is not divisible by 5 as the remainder is 3.
n+6 = 5q+6 = (5q+5)+1, is not divisible by 5 as the remainder is 1.
n+12 = 5q+12 = (5q+10)+2, is not divisible by 5 as the remainder is 2.

Therefore, in this case only one out of n, n+1, n+8, n+6, n+12 is divisible by 5 which is n.
When,
 n = 5q+1, n is not divisible by 5 as the remainder is 1.
n+4 = 5q+1+4 = 5q+5, is divisible by 5.
n+8 = 5q+1+8 = (5q+5)+4, is not divisible by 5 as the remainder is 4.
n+6 = 5q+1+6 = (5q+5)+2, is not divisible by 5 as the remainder is 2.
n+12 = 5q+1+12 = (5q+10)+3, is not divisible by 5 as the remainder is 3.

Therefore, in this case only one out of n, n+4, n+8, n+6, n+12 is divisible by 5 which is n+4.
When,
 n = 5q+2, n is not divisible by 5 as the remainder is 2.
n+4 = 5q+2+4 = (5q+5)+1, is not divisible by 5 as the remainder is 1.
n+8 = 5q+2+8 = 5q+10, is divisible by 5 as the remainder is 0.
n+6 = 5q+2+6 = (5q+5)+3, is not divisible by 5 as the remainder is 3.
n+12 = 5q+2+12 = (5q+10)+4, is not divisible by 5 as the remainder is 4.

Therefore, in this case only one out of n, n+4, n+8, n+6, n+12 is divisible by 5 which is n+8.
When,
 n = 5q+3, n is not divisible by 5 as the remainder is 3.
n+4 = 5q+3+4 = (5q+5)+2, is not divisible by 5 as the remainder is 2.
n+8 = 5q+3+8 = (5q+10)+1, is not divisible by 5 as the remainder is 1.
n+6 = 5q+3+6 = (5q+5)+4, is not divisible by 5 as the remainder is 4.
n+12 = 5q+3+12 = 5q+15, is divisible by 5 as the remainder is 0.

Therefore, in this case only one out of n, n+4, n+8, n+6, n+12 is divisible by 5 which is n+12.
When,
 n = 5q+4, n is not divisible by 5 as the remainder is 4.
n+4 = 5q+4+4 = (5q+5)+3, is not divisible by 5 as the remainder is 3.
n+8 = 5q+4+8 = (5q+10)+2, is not divisible by 5 as the remainder is 2.
n+6 = 5q+4+6 = 5q+10, is divisible by 5 as the remainder is 0.
n+12 = 5q+4+12 = (5q+15)+1, is not divisible by 5 as the remainder is 1.
Therefore, in this case only one out of n, n+4, n+8, n+6, n+12 is divisible by 5 which is n+6.
Hence, one and only one out of n, n+4, n+8, n+6 and n+12 is divisible by 5.

Note: Here we need to check all the 5 numbers for each value of r. If we check only of r = 0 or r = 1, we will not be able to conclude that the result holds.