If n > 5, find the remainder when 7n – 6n(3n – 2) – 7 is divided by 216.431210Cannot be determined

1. Break Down the Problem

We need to find the remainder when the expression $7n - 6n(3n - 2) - 7$ is divided by 216 for $n > 5$.

2. Relevant Concepts

We'll simplify the expression $7n - 6n(3n - 2) - 7$ first, and then we'll perform the division.

3. Analysis and Detail

Let's simplify the expression step by step:

1. Distribute $6n$ in $6n(3n - 2)$: $6n(3n - 2) = 18n^2 - 12n$

2. Substitute this back into the original expression: $7n - (18n^2 - 12n) - 7$

3. Combine like terms: $= 7n - 18n^2 + 12n - 7$ $= -18n^2 + 19n - 7$

4. Verify and Summarize

Now we have the simplified expression $-18n^2 + 19n - 7$. We will divide this expression by 216 and find the remainder.

Assuming $n = 6$ (the smallest integer greater than 5): $-18(6^2) + 19(6) - 7$ $= -18(36) + 114 - 7$ $= -648 + 114 - 7$ $= -541$

Now we need to find the remainder of $-541$ when divided by $216$: Calculating $-541 \div 216$: $-541 = -3 \times 216 + 107$

Thus: $\text{Remainder} = 107$

Final Answer

The remainder when $7n - 6n(3n - 2) - 7$ is divided by 216, when $n > 5$, is $\boxed{107}$.