Question 36

Show that one and only one out of $n,n+2\text{or}n+4$ is divisible by 3, where $n$ is any positive integer.

Solution:

To show that one and only one out of $$n,n+2,n+4$$ is divisible by 3, where $$n$$ is any positive integer, we can analyze the possible cases based on the value of $$n$$ modulo 3.

1. Consider the possible values of $$n$$ modulo 3:
– Any integer $$n$$ can be expressed in one of three forms based on its remainder when divided by 3:
– Case 1: $$n\equiv 0\mathrm{mod}3$$
– Case 2: $$n\equiv 1\mathrm{mod}3$$
– Case 3: $$n\equiv 2\mathrm{mod}3$$

2. Analyze Case 1: $$n\equiv 0\mathrm{mod}3$$:
– If $$n\equiv 0\mathrm{mod}3$$, then:
– $$n$$ is divisible by 3.
– $$n+2\equiv 2\mathrm{mod}3$$ (not divisible by 3)
– $$n+4\equiv 1\mathrm{mod}3$$ (not divisible by 3)
– Conclusion: In this case, only $$n$$ is divisible by 3.

3. Analyze Case 2: $$n\equiv 1\mathrm{mod}3$$:
– If $$n\equiv 1\mathrm{mod}3$$, then:
– $$n$$ is not divisible by 3.
– $$n+2\equiv 0\mathrm{mod}3$$ (divisible by 3)
– $$n+4\equiv 1\mathrm{mod}3$$ (not divisible by 3)
– Conclusion: In this case, only $$n+2$$ is divisible by 3.

4. Analyze Case 3: $$n\equiv 2\mathrm{mod}3$$:
– If $$n\equiv 2\mathrm{mod}3$$, then:
– $$n$$ is not divisible by 3.
– $$n+2\equiv 1\mathrm{mod}3$$ (not divisible by 3)
– $$n+4\equiv 0\mathrm{mod}3$$ (divisible by 3)
– Conclusion: In this case, only $$n+4$$ is divisible by 3.

5. Final Conclusion:
– In all three cases, we have shown that one and only one of the numbers $$n,n+2,n+4$$ is divisible by 3.