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1. True. 4 “goes into” 20 five times without remainder. In other words, \(20 \div 4 = 5\text{,}\) an integer. We could also justify this by saying that \(20\) is a multiple of 4: \(20 = 4\cdot 5\text{.}\)

2. False. While 20 is a multiple of 4, it is false that \(4\) is a multiple of 20.

3. False. \(5 \div 0\) is not even defined, let alone an integer.

4. True. In fact, \(x \mid 0\) is true for all \(x\text{.}\) This is because 0 is a multiple of every number: \(0 = x\cdot 0\text{.}\)

5. True. In fact, \(x \mid x\) is true for all \(x\text{.}\)

6. True. 1 divides every number (other than 0).

7. True. Negative numbers work just fine for the divisibility relation. Here \(12 = -3 \cdot 4\text{.}\) It is also true that \(3 \mid -12\) and that \(-3 \mid -12\text{.}\)

8. False. Both 8 and 12 are divisible by 4, but this does not mean that \(12\) is divisible by \(8\text{.}\)

9. False. See below.