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### 1. **Is it easier to find all multiples or all factors of a number?**
It is generally easier to find all factors of a number, especially smaller numbers, because factors are finite, while multiples are infinite. For example:

- **Factors of 12**: \(1, 2, 3, 4, 6, 12\). These are all the numbers that divide 12 evenly. There are only six factors.
- **Multiples of 12**: \(12, 24, 36, 48, 60, \dots\). Multiples continue infinitely, which makes finding all multiples much harder.

Since multiples continue without end, you can only list a portion of them, while the number of factors is always limited and can be found using a systematic approach.

#### Example of testing:
- **Factors of 15**: \(1, 3, 5, 15\)
- **Multiples of 15**: \(15, 30, 45, 60, 75, \dots\)
  
Finding all multiples would require infinite work, while finding factors stops at the largest number itself.

### 2. **Two numbers less than 10 that cannot possibly be common factors of 40 and 80**
The common factors of 40 and 80 are the numbers that divide both exactly. We know:

- **Factors of 40**: \(1, 2, 4, 5, 8, 10, 20, 40\)
- **Factors of 80**: \(1, 2, 4, 5, 8, 10, 20, 40, 80\)

Two numbers less than 10 that could not possibly be common factors of 40 and 80 are **3** and **7** because they do not divide either 40 or 80.

### 3. **How to show that 45 is a common factor of 450 and 315**
To show that 45 is a common factor of 450 and 315, we can divide both numbers by 45 and check if the result is an integer:

- \( \frac{450}{45} = 10 \) (an integer)
- \( \frac{315}{45} = 7 \) (an integer)

Since 45 divides both 450 and 315 exactly, it is a common factor.

### 4. **How to show that 90 is a common multiple of 6 and 45**
To show that 90 is a common multiple of 6 and 45, we check if 90 can be divided by both 6 and 45:

- \( \frac{90}{6} = 15 \) (an integer)
- \( \frac{90}{45} = 2 \) (an integer)

Since 90 can be divided by both 6 and 45 without a remainder, it is a common multiple of both numbers.

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### Do you have any questions or want more details?

Here are five related questions you could explore further:
1. What are the factors of 60 and how do they compare to 40?
2. Can you find a number that is a common factor of 60 and 45?
3. What is the least common multiple (LCM) of 12 and 18?
4. How do prime factors help in finding common factors of two numbers?
5. How can you quickly find multiples of a number up to 100?

**Tip**: When finding common factors, prime factorization can be a very useful tool!

Do you think it’s easier to find all multiples of a number or all factors of a number? Show and explain your thinking using specific examples. List two numbers less than 10 that could not possibly be common factors of 40 and 80. How could you show why 45 is a common factor of 450 and 315? How could you show that 90 is a common multiple of 6 and 45?

Explore whether it’s easier to find factors or multiples of a number with detailed examples. Learn how to identify common factors and multiples, including why 45 is a common factor of 450 and 315, and how 90 is a common multiple of 6 and 45. Suitable for Grades 5-7.

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