For a, b ∈ Z and n ∈ N define a ̸≡ b (mod n) to mean a is not

congruent to b modulo n that is a − b is not a multiple of n. Write the

contrapositive and then prove or disprove the following statement.

For a, b ∈ Z, if a ̸≡ 0 (mod 10) and b ̸≡ 0 (mod 10), then ab ̸≡ 0 (mod 10)

The statement is:

> For \( a, b \in \mathbb{Z} \), if \( a \not\equiv 0 \pmod{10} \) and \( b \not\equiv 0 \pmod{10} \), then \( ab \not\equiv 0 \pmod{10} \).

Let’s break down and analyze this by following the steps outlined:

### Step 1: Write the Contrapositive

To write the contrapositive, we negate both the hypothesis and the conclusion and switch their positions. Thus, the contrapositive of the statement is:

> If \( ab \equiv 0 \pmod{10} \), then \( a \equiv 0 \pmod{10} \) or \( b \equiv 0 \pmod{10} \).

This means that if the product \( ab \) is divisible by 10, then at least one of \( a \) or \( b \) must be divisible by 10.

### Step 2: Prove or Disprove the Statement

The contrapositive is often useful in proving an implication, as proving the contrapositive is logically equivalent to proving the original statement.

#### Proof of the Contrapositive

Suppose \( ab \equiv 0 \pmod{10} \). This means that \( ab \) is a multiple of 10.

For \( ab \) to be a multiple of 10, at least one of \( a \) or \( b \) must be divisible by either 2 or 5 (since 10 has the prime factors 2 and 5). However, for \( ab \) to be divisible by 10 specifically, either \( a \) or \( b \) must include both 2 and 5 as factors, meaning that at least one of \( a \) or \( b \) must be a multiple of 10.

Thus, if \( ab \equiv 0 \pmod{10} \), then at least one of \( a \) or \( b \) must be congruent to 0 modulo 10. This completes the proof of the contrapositive.

### Conclusion

Since we have proven the contrapositive, the original statement is **true**:

> For \( a, b \in \mathbb{Z} \), if \( a \not\equiv 0 \pmod{10} \) and \( b \not\equiv 0 \pmod{10} \), then \( ab \not\equiv 0 \pmod{10} \).

Would you like further clarification on any part of this proof?

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Here are five follow-up questions for deeper exploration:

1. What is the significance of using the contrapositive in proofs?
2. Can this result be extended to other moduli beyond 10? Why or why not?
3. What are other methods to prove statements about divisibility and congruences?
4. How does this proof change if we use different primes instead of 2 and 5?
5. What are examples of numbers \( a \) and \( b \) such that \( ab \not\equiv 0 \pmod{10} \) even though \( a \) and \( b \) are not congruent to 0 modulo 10?

**Tip:** When proving statements in modular arithmetic, consider trying the contrapositive approach—it can simplify the proof by focusing on easier or more intuitive implications.

For a, b ∈ Z and n ∈ N define a ̸≡ b (mod n) to mean a is not congruent to b modulo n that is a − b is not a multiple of n. Write the contrapositive and then prove or disprove the following statement: For a, b ∈ Z, if a ̸≡ 0 (mod 10) and b ̸≡ 0 (mod 10), then ab ̸≡ 0 (mod 10).

Explore a proof involving modular arithmetic and contrapositive logic. Learn how to prove that if a ≢ 0 (mod 10) and b ≢ 0 (mod 10), then ab ≢ 0 (mod 10). This solution includes step-by-step reasoning suitable for college-level students.

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