1. **State the problem:** Prove that for any non-negative integer $n$, the expression $32n + 7$ is divisible by 8. 2. **Recall divisibility rules and formulas:** A number $a$ is divisible by $b$ if $a = b \times k$ for some integer $k$. 3. **Analyze the expression:** Consider $32n + 7$. 4. **Check divisibility by 8:** Since $32 = 8 \times 4$, rewrite the expression as: $$32n + 7 = 8 \times 4n + 7$$ 5. **Evaluate modulo 8:** Taking modulo 8 on both terms, $$32n \equiv 0 \pmod{8}$$ because $32n$ is a multiple of 8. 6. However, $$7 \equiv 7 \pmod{8}$$ 7. Adding these, $$32n + 7 \equiv 0 + 7 = 7 \pmod{8}$$ 8. Since the remainder is 7, not 0, $32n + 7$ is **not divisible** by 8 for any integer $n$. **Final conclusion:** The statement is false; $32n + 7$ is not divisible by 8 for any non-negative integer $n$.