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The problem asks us to find the largest possible number that Alexander could be thinking of, given that 20 and 32 are multiples of this number. 

To solve this, we need to find the **Greatest Common Divisor (GCD)** of 20 and 32 because the largest number that divides both 20 and 32 is the GCD.

### Steps:
1. **Prime factorization of 20:**
   \[
   20 = 2^2 \times 5
   \]

2. **Prime factorization of 32:**
   \[
   32 = 2^5
   \]

3. The common factor between 20 and 32 is \(2\), and the smallest power of 2 in both factorizations is \(2^2\).

4. Therefore, the **GCD** of 20 and 32 is \(2^2 = 4\).

### Conclusion:
The largest possible number Alexander could be thinking of is **4**.

Would you like further details on any of these steps?

Here are five questions that expand on this concept:
1. How do you compute the GCD of more than two numbers?
2. Why is the GCD useful in problem-solving?
3. What are other methods to find the GCD besides prime factorization?
4. How would this problem change if 20 and 32 were not multiples of the same number?
5. Can the GCD be larger than the smallest of the given numbers?

**Tip:** Always double-check your prime factorizations, as small mistakes can lead to incorrect GCD calculations.

Alexander thinks of a number. 20 and 32 are multiples of his number. Find the largest possible number he could be thinking of.

Learn how to find the largest possible number that divides both 20 and 32. This problem involves concepts of number theory, including prime factorization and the Greatest Common Divisor (GCD). Suitable for students in Grades 5-8.

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