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For any integer n, prove that
1) 3 divides one of the integers n, n + 1 or 2n + 1.
2) 3 divides one of n, n + 2 or n + 4.
3) 3 divides one of n, 2n - 1 or 2n +1.
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1) 3 divides one of the integers n, n + 1 or 2n + 1.
a) If n divides 3 then the statement is proved.
If n does not divide 3 without a remainder, then the remainder is either 1 or 2.
b) If the remainder is 1, then 2n+1 has the remainder 2*1+1 = 3;
in other words, then 2n +1 is a multiple of 3.
c) If the remainder is 2, then n+1 is divisible by 3.
Thus, in all cases one of the three numbers n, n+1 or 2n+1 is divisible by 3.
Proved.
2) 3 divides one of n, n + 2 or n + 4.
a) If n divides 3 then the statement is proved.
If n does not divide 3 without a remainder, then the remainder is either 1 or 2.
b) If the remainder is 1, then n+2 is divisible by 3.
c) If the remainder is 2, then n+4 is divisible by 3.
Thus, in all cases one of the three numbers n, n+2 or n+4 is divisible by 3.
Proved.
Try to prove the statement #3 by the same method. (Same logic, same arguments).
Good luck !!