Let 𝑎 and 𝑏 be integers and 𝑛 be a natural number. Prove:
If 10𝑎 ≢ 10𝑏 (mod 𝑛), then 𝑛 ∤ 10. Use the contrapositive.

To prove the statement:  
"If \(10a \not\equiv 10b \pmod{n}\), then \(n \nmid 10\),"  
we will use the **contrapositive** approach. The contrapositive of a statement "if \(P\), then \(Q\)" is logically equivalent to "if not \(Q\), then not \(P\)". Thus, instead of proving the original statement directly, we will prove the contrapositive, which is:

**Contrapositive:**  
If \(n \mid 10\), then \(10a \equiv 10b \pmod{n}\).

### Proof:

#### 1. Assume \(n \mid 10\):
This means that \(n\) divides \(10\), i.e., \(n = 1\), \(n = 2\), \(n = 5\), or \(n = 10\) (since 10 has divisors 1, 2, 5, and 10).

#### 2. Consider \(10a \pmod{n}\) and \(10b \pmod{n}\):
If \(n \mid 10\), then for any integer values of \(a\) and \(b\), the expressions \(10a\) and \(10b\) both become multiples of \(n\). In other words, \(10a\) and \(10b\) are congruent modulo \(n\) because \(n\) divides 10, and hence \(n\) divides the difference \(10a - 10b\).

- Specifically, since \(10a \equiv 0 \pmod{n}\) and \(10b \equiv 0 \pmod{n}\), we have:
  \[
  10a \equiv 10b \pmod{n}.
  \]

#### 3. Conclusion:
Thus, if \(n \mid 10\), it follows that \(10a \equiv 10b \pmod{n}\) for any integers \(a\) and \(b\).

Since we have proven the contrapositive, it logically follows that the original statement is true:
"If \(10a \not\equiv 10b \pmod{n}\), then \(n \nmid 10\)."

### Final Conclusion:
The statement is proven using the contrapositive.

Would you like further details on this proof or examples to clarify any part?

Here are five related questions for you to explore further:

1. How do you prove statements using direct proofs rather than contrapositive?
2. What is the importance of contrapositive in mathematical logic?
3. What are other examples of statements that can be proved by contrapositive?
4. How does modular arithmetic behave under multiplication?
5. What is the relationship between divisibility and modular arithmetic?

**Tip:** Always remember that proving a contrapositive is logically equivalent to proving the original statement!

Let 𝑎 and 𝑏 be integers and 𝑛 be a natural number. Prove: If 10𝑎 ≢ 10𝑏 (mod 𝑛), then 𝑛 ∤ 10. Use the contrapositive.

This problem demonstrates how to use a contrapositive proof to show that if 10a is not congruent to 10b mod n, then n does not divide 10. Explore key number theory concepts like modular arithmetic and divisibility rules, suitable for students in Grades 10-12.

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