1st..consider a=3..then there r 3 possibilities..3m,3m+1or 3m+2..then put this on d place of n.. then u can prove it..

Introduction:

To prove that exactly one of the numbers n, n^2, or n^4 is divisible by 3, we need to consider all possible cases and provide a logical explanation for each one.

Case 1: n is divisible by 3
If n is divisible by 3, then n^2 and n^4 will also be divisible by 3, as any number raised to a power greater than or equal to 1 will retain the divisibility properties of the original number.

Case 2: n is not divisible by 3 and leaves a remainder of 1 when divided by 3
If n is not divisible by 3 and leaves a remainder of 1 when divided by 3, then n^2 will also leave a remainder of 1 when divided by 3. This is because when the original number has a remainder of 1, its square will also have a remainder of 1 when divided by 3. However, n^4 will be divisible by 3 in this case. This is because (n^2)^2 is equal to n^4, and if n^2 leaves a remainder of 1 when divided by 3, then its square, n^4, will be divisible by 3.

Case 3: n is not divisible by 3 and leaves a remainder of 2 when divided by 3
If n is not divisible by 3 and leaves a remainder of 2 when divided by 3, then n^2 will leave a remainder of 1 when divided by 3. This is because when the original number has a remainder of 2, its square will have a remainder of 1 when divided by 3. However, n^4 will not be divisible by 3 in this case. This is because (n^2)^2 is equal to n^4, and if n^2 leaves a remainder of 1 when divided by 3, then its square, n^4, will also leave a remainder of 1 when divided by 3.

Conclusion:

In conclusion, we have examined all possible cases for the number n. In case 1, all three numbers n, n^2, and n^4 are divisible by 3. In case 2, n and n^2 are not divisible by 3, but n^4 is divisible by 3. In case 3, n^2 is not divisible by 3, but n and n^4 are not divisible by 3. Therefore, in each case, exactly one of the numbers n, n^2, or n^4 is divisible by 3.