2.4 Find Multiples and Factors

Understanding Multiples and Factors

Multiples and factors describe how numbers relate to each other through multiplication and division. They're the foundation for simplifying fractions, finding common denominators, and working with larger numbers later on.

Patterns of number multiples

A multiple of a number is the product you get when you multiply that number by any counting number (1, 2, 3, 4, ...).

• Multiples of 3: 3, 6, 9, 12, 15, 18, ...
• Multiples of 7: 7, 14, 21, 28, 35, 42, ...

There are a few handy patterns to notice:

• Even number multiples are always even. For example, every multiple of 4 is even: 4, 8, 12, 16, ...
• Odd number multiples alternate between odd and even. Look at the multiples of 7: 7 (odd), 14 (even), 21 (odd), 28 (even), ...
• Multiples of 5 always end in 0 or 5: 5, 10, 15, 20, 25, ...
• Multiples of 10 always end in 0: 10, 20, 30, 40, ...

Divisibility rules for 2, 3, 5, and 10

Divisibility rules are shortcuts that tell you whether one number divides evenly into another without actually doing the division.

• Divisible by 2: The last digit is even (0, 2, 4, 6, or 8).
  • 234 is divisible by 2 because the last digit, 4, is even.
• Divisible by 3: Add up all the digits. If that sum is divisible by 3, so is the original number.
  • 135 → $1 + 3 + 5 = 9$, and 9 is divisible by 3, so 135 is too.
• Divisible by 5: The last digit is 0 or 5.
  • 275 and 480 are both divisible by 5.
• Divisible by 10: The last digit is 0.
  • 350 and 1,240 are both divisible by 10.

These rules help you identify factors quickly, which you'll use constantly when simplifying fractions.

Factors and Number Classifications

Systematic factor identification

A factor of a number is any whole number that divides into it evenly (with no remainder). For example, the factors of 18 are 1, 2, 3, 6, 9, and 18.

The most reliable way to find all factors of a number is to work through factor pairs. Here's how:

1. Start with 1. Since $1 \times 36 = 36$, the pair is 1 and 36.
2. Try 2. Since $2 \times 18 = 36$, the pair is 2 and 18.
3. Try 3. Since $3 \times 12 = 36$, the pair is 3 and 12.
4. Try 4. Since $4 \times 9 = 36$, the pair is 4 and 9.
5. Try 5. $36 \div 5 = 7.2$, which isn't a whole number, so 5 is not a factor.
6. Try 6. Since $6 \times 6 = 36$, the pair is 6 and 6.

You can stop checking once the two numbers in a pair meet or cross over (here, at 6). That means you've found every factor. The complete list for 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Prime vs. composite numbers

Once you understand factors, you can classify any whole number greater than 1:

• A prime number has exactly two factors: 1 and itself. It can't be broken into a product of smaller whole numbers.
  • Examples: 2, 3, 5, 7, 11, 13, 17, 19
  • Notice that 2 is the only even prime number. Every other even number is divisible by 2, so it has at least three factors.
• A composite number has more than two factors, meaning it can be broken into a product of smaller numbers.
  • Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16

Number theory concepts

Number theory is the branch of math focused on the properties of integers (whole numbers, including positives, negatives, and zero). The ideas in this section, like factors, multiples, divisibility, and primes, are all part of number theory. You'll keep building on these concepts as you move into topics like greatest common factor and least common multiple.