To prove the divisibility of \( 8^{10} - 8^9 - 8^8 \) by 55, we can start by simplifying the expression.
First, we factor out the common term \( 8^8 \):
\[ 8^{10} - 8^9 - 8^8 = 8^8 (8^2 - 8 - 1) \]
Next, we need to simplify the expression inside the parentheses:
\[ 8^2 - 8 - 1 = 64 - 8 - 1 = 55 \]
So we have:
\[ 8^{10} - 8^9 - 8^8 = 8^8 \cdot 55 \]
Since this expression is the product of \( 8^8 \) and 55, it is clear that \( 8^{10} - 8^9 - 8^8 \) is divisible by 55.
Thus, we can conclude that:
\[ 8^{10} - 8^9 - 8^8 \text{ is divisible by } 55. \]
1 answer