Show that one and only one out of n, (n + 2) and (n + 4) is divisible by 3, where n is any positive integer.
We applied Euclid Division algorithm on n and 3.
a = bq + r on putting a = n and b = 3
n = 3q + r, 0 < r < 3
So,
n = 3q (I)
n = 3q + 1 (II)
n = 3q + 2 (III)
Case - I: When n = 3q
In this case, we have
n = 3q, which is divisible by 3
Now, n = 3q
n + 2 = 3q + 2
n + 2 leaves remainder 2 when divided by 3
Again, n = 3q
n + 4 = 3q + 4 = 3(q + 1) + 1
n + 4 leaves remainder 1 when divided by 3
n + 4 is not divisible by 3.
Thus, n is divisible by 3 but n + 2 and n + 4 are not divisible by 3.
Case - II: when n = 3q + 1
In this case, we have
n = 3q + 1,
n leaves remainder 1 when divided by 3.
n is divisible by 3
Now, n = 3q + 1
n + 2 = (3q + 1) + 2 = 3(q + 1)
n + 2 is divisible by 3.
Again, n = 3q + 1
n + 4 = 3q + 1 + 4 = 3q + 5 = 3(q + 1) + 2
n + 4 leaves remainder 2 when divided by 3
n + 4 is not divisible by 3.
Thus, n + 2 is divisible by 3 but n and n + 4 are not divisible by 3.
Case - III: When n + 3q + 2
In this case, we have
n = 3q + 2
n leaves remainder 2 when divided by 3.
n is not divisible by 3.
Now, n = 3q + 2
n + 2 = 3q + 2 + 2 = 3(q + 1) + 1
n + 2 leaves remainder 1 when divided by 3
n + 2 is not divisible by 3.
Again, n = 3q + 2
n + 4 = 3q + 2 + 4 = 3(q + 2)
n + 4 is divisible by 3.
Thus, n + 4 is divisible by 3 but n and n + 2 are not divisible by 3.