Question

Show that one and only one out of n, n+2 or n +4 is divisible by 3, where n is any positive integer​

Answer 1

Answer:

We know that if we divide a whole number with 3, the remainders we get are 0,1 or  2 only. For the conditions, we will be assuming that the remainder is given when N is divided by 3

Case I : Remainder = 0

If N/3 gives remainder 0, it means that N is divisible by 3

Case II : Remainder = 1

If N/3 gives remainder 1, (N+1)/3 would give a remainder of 2, and hence (N+2)/3 will give a remainder of 0. Therefore, N+2 is divisible

Case III : Remainder = 3

If N/3 gives remainder 2, (N+1)/3 would give a remainder of 0, and hence (N+2)/3 will give a remainder of 1. And so (N+2) is not divisible. And we continue

If (N+2)/3 gives remainder 1, (N+3)/3 would give a remainder of 2, and hence (N+4)/3 will give a remainder of 0. Therefore, N+4 is divisible by 3

We find that in all 3 cases, either one of N, (N+2) or (N+4) is divisible by 3. Hence proved

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