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Answer
An integer with factors 2 and 5 is 10.
Solution
To find an integer that has factors 2 and 5, we can follow these steps:
1. **Understanding Factors**: If an integer has factors 2 and 5, it means that the integer can be expressed as a product of these factors. Specifically, the integer must be a multiple of both 2 and 5.
2. **Finding the Least Common Multiple (LCM)**: The least common multiple of 2 and 5 will give us the smallest integer that has both as factors. The LCM of two numbers is the smallest number that is a multiple of both.
3. **Calculating the LCM**:
- The prime factorization of 2 is \(2^1\).
- The prime factorization of 5 is \(5^1\).
- The LCM is found by taking the highest power of each prime factor:
\[
\text{LCM}(2, 5) = 2^1 \times 5^1 = 10
\]
4. **Conclusion**: The smallest integer that has both 2 and 5 as factors is 10. However, any multiple of 10 (like 20, 30, 40, etc.) will also have 2 and 5 as factors.
Thus, an integer with factors 2 and 5 is 10, and any integer of the form \(10n\) where \(n\) is an integer will also satisfy this condition.
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Did you know that any integer with factors 2 and 5 is guaranteed to be a multiple of 10? This is because the product of 2 and 5 equals 10. So, if you've got an integer that can be divided evenly by both 2 and 5, you can always count on it being part of the exclusive club of multiples of 10, meaning it ends with a 0! When working with numbers that have specific factors, a common mistake is to forget about additional conditions like the presence of other factors. For instance, just because a number is divisible by 2 and 5 doesn’t mean it can't be divisible by other numbers like 3 or 7! So, always consider the full landscape of the number's factors when you're diving into its properties.
