Show that p-1 is a factor of `p^(10)-1 `and also of `p^(11)-1.`

Explanation

To show that \( p - 1 \) is a factor of \( p^{10} - 1 \) and \( p^{11} - 1 \), we can use the Remainder Theorem. According to this theorem, if a polynomial \( f(p) \) is divided by \( p - a \), the remainder is \( f(a) \). If the remainder is 0, then \( p - a \) is a factor of \( f(p) \).

### Step-by-Step Solution:

**Step 1: Show that \( p - 1 \) is a factor of \( p^{10} - 1 \)**

1. Let \( f(p) = p^{10} - 1 \).
2. We want to check if \( p - 1 \) is a factor of \( f(p) \). According to the Remainder Theorem, we need to evaluate \( f(1) \):
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