## What Are Factors and Multiples?

**Factors**** and ****multiples**** are two interrelated concepts in mathematics. If **$A \times B = C$**, then A and B are factors of C, whereas C is a multiple of both A and B.**

Consider an example to understand.

**Example:** $3 \times 7 = 21$

Here, 3 and 7 are factors of 21.

21 is the multiple of both 3 and 7.

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## What Are Factors?

A factor is a number that divides the given number exactly, without any remainder. When a number n is divisible by b, we can say that b is a factor of n.

For example, the factors of 16 are 1, 2, 4, 8, and 16, because these numbers divide 16 exactly, without leaving any remainder.

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## Properties of Factors

- 1 is the smallest factor of any number.
- The number itself is the greatest factor of any number.
- A factor of a number is always less than or equal to the number.
- Factors of a number are finite.
- 0 cannot be a factor of any number.
- If B is a multiple of A, then A is a factor of B.

## What Are Multiples?

A multiple of a number is the number obtained when the given number is multiplied with an integer.

A multiple is a result of multiplying a number by an integer. Note that, when we study multiples of any number, we generally talk about positive multiples only (excluding 0 and negative multiples). Thus, we can say that a multiple is the number obtained by multiplying a given number by a positive integer.

For example, the multiples of 5 are 5, 10, 15, 20,…, and so on, because these numbers can be obtained by multiplying 5 with the whole numbers 1, 2, 3, 4,…, and so on.

You can find the multiples of a number in the multiplication table (times table). Take a look!

$5 \times 1 = 5$

$5 \times 2 = 10$

$5 \times 3 = 15$

$5 \times 4 = 20$

$5 \times 5 = 25$

$5 \times 6 = 30$

$5 \times 7 = 35$

$5 \times 8 = 40$

$5 \times 9 = 45$

$5 \times 10 = 50$

## Properties of Multiples

- Infinitely many multiples of any number.
- A multiple is always greater than or equal to the given number.
- Every number is a multiple of 1.
- If B is a multiple of A, then A is a factor of B.
- 0 is considered a multiple of every number.

## Difference Between Factors and Multiples

Factors | Multiples |
---|---|

Factors are numbers that divide the given number without leaving a remainder. | Multiples are the numbers obtained as a product of a given number and integers. |

A factor of a number is less than or equal to itself. | Multiples are always greater than or equal to the given number. |

Factors are finite in number and include 1 and the number itself. | There are an infinite number of multiples for a given number. |

1 is the smallest factor of any number. | Every number is a multiple of 1. |

0 cannot be a factor of any number. | 0 is a multiple of every number. |

Example: Factors of 8 are 1, 2, 4, 8. | Example: Multiples of 8 are 8, 16, 24,… |

## How to Find Factors and Multiples

Let’s move ahead and learn the methods we can use to find factors and multiples of a number.

**How can we find factors?**

To find the factors of a given number, divide the number by each integer between 1 and the number itself. The integers that give us a remainder of 0 are factors of the number.

For example, to find the factors of 6, divide 6 by numbers 1 to 6.

61 | Remainder $= 0$ |

62 | Remainder $= 0$ |

63 | Remainder $= 0$ |

64 | Remainder $= 0$ |

65 | Remainder $= 1$ |

66 | Remainder $= 0$ |

The factors of 6 are 1, 2, 3, and 6.

**How to find multiples**

To find the multiples of a given number, multiply the number by each positive integer.

For example, to find the multiples of 5, you would multiply 5 by 1, 2, 3, 4, and so on.

Multiples of $5 = 5,\; 10,\; 15,\; 20,\; 25$, …

Multiples of $9 = 9\;, 18\;, 27\;, 36\;, 45$, …

## Least Common Multiple (LCM)

LCM of two numbers is the smallest positive integer that is a multiple of both the numbers. It is the smallest number among the common multiples of the given two numbers.

Let’s find the LCM of 4 and 6.

Multiples of 4 | 4 | 8 | 12 | 16 | 20 | 24 | … |

Multiples of 6 | 6 | 12 | 18 | 24 | 30 | 36 | … |

Common multiples $= 12,\; 24,\; 36$, …

Least common multiple $= 12$

LCM (4, 6) $= 12$

## Greatest Common Factor (GCF)

It is the largest positive integer that divides the given two integers without leaving any remainder. In other words, it is the largest number that is a factor of all the given integers.

Let’s find the GCD of 12 and 18.

Factors of 12 | 1 | 2 | 3 | 6 | 12 | – |

Factors of 18 | 1 | 2 | 3 | 6 | 9 | 18 |

The largest number that appears in both lists is 6.

GCF of 12 and 18 is 6.

## Facts about Factors and Multiples

- Every number has at least two factors, which are 1 and itself.
- A number can have an infinite number of multiples, but only a finite number of factors.
- Factors and multiples can be used to find the patterns and symmetries in nature, such as the arrangement of petals in flowers and the formation of crystals.
- Prime numbers have only two factors, 1 and itself, while composite numbers have more than two factors.
- The number 1 is neither prime nor composite. It only has one factor, which is itself.
- 0 is neither prime nor composite.

## Conclusion

In this article, we have learned about factors and multiples, the definitions, how to find factors and multiples, and also the properties of multiples and factors of a number. We have also learned some interesting facts about factors and multiples. Let’s now look at some solved examples and practice problems to understand the concept better.

## Solved Examples on Factors and Multiples

**Find the factors of 30.**

**Solution: **

The factors of 30 are the integers that divide 30 without leaving any remainder.

$30 \div 1 = 30 \text{and remainder} = 0$

$30 \div 2 = 15 \text{and remainder} = 0$

$30 \div 3 = 10 \text{and remainder} = 0$

$30 \div 5 = 6 \text{and remainder} = 0$

$30 \div 6 = 5 \text{and remainder} = 0$

30 \div 10 = 3 \text{and remainder} = 0$

$30 \div 15 = 2 \text{and remainder} = 0$

$30 \div 30 = 1 \text{and remainder} = 0$

Thus, the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.

**Find the multiples of 7 that are less than or equal to 56.**

**Solution: **

We can write the multiples of 7 as,

$7 \times 1 = 7$

$7 \times 2 = 14$

$7 \times 3 = 21$

$7 \times 4 = 28$

$7 \times 5 = 35$

$7 \times 6 = 42$

$7 \times 7 = 49$

$7 \times 8 = 56$

$7 \times 9 = 63$

Hence, the multiples of 7 that are less than or equal to 56 are 7, 14, 21, 28, 35, 42, 49, and 56.

**Find the least common multiple (LCM) of 8 and 12.**

**Solution: **

Multiples of 8: 8, 16, 24, 32, 40, 48…

Multiples of 12: 12, 24, 36, 48…

Common multiples: 24, 48…

LCM of 8 and 12 is the smallest number that is a multiple of both 8 and 12, which is 24.

**Find the greatest common factor (GCF) of 18 and 48.**

**Solution: **

Factors of 18 are 1, 2, 3, 6, 9, and 18.

Factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Hence, the GCF of 18 and 48 is the greatest number that is a factor of both 18 and 24, which is 6.

**If 20 and 35 are both factors of a number n, what is the smallest possible value of n?**

**Solution: **

The smallest possible value of n is the least common multiple (LCM) of 20 and 35.

The multiples of 20 are 20, 40, 60, 80, 100, 120, **140**, 160, 180… and so on.

And

The multiples of 35 are 20, 40, 60, 80, 100, 120, **140**, 160, 180… and so on.

LCM of 20 and $35 = 140$.

Hence, the smallest possible value of n is 140.

## Practice Problems on Factors and Multiples

## Factors and Multiples - Definition, Differences, Examples, FAQs

### Which of the following is a factor of 27?

A factor of a number is a whole number that divides evenly into that number. In this case, 9 divides 27 evenly, whereas 2, 5, and 16 do not.

### What is the greatest common factor of 12 and 36?

The factors of 24 are 1, 2, 3, 4, 6, 8, and 12.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12.

The largest factor that they have in common is 12.

### Which of the following numbers is a multiple of 5?

15 is a multiple of 5 because it is obtained by multiplying 5 by 3.

### The LCM of 4 and 7 is ______.

The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36...

the multiples of 7 are 7, 14, 21, 28, 35…

Common multiples: 28, 56, …

Hence, the LCM of both 4 and 7 is 28.

### What is the sum of the first five multiples of 3?

The first five multiples of 3 are 3, 6, 9, 12, and 15.

$3 + 6 + 9 + 12 + 15 = 45$.

## Frequently Asked Questions about Factors and Multiples

**What are prime factors?**

Prime factors are the factors of a number that are prime numbers. For example, the prime factors of 6 are 2 and 3 since 6 can be written as $2 \times 3$.

**What is the difference between factors and multiples?**

A factor of a number is a number that divides it evenly, while a multiple of a number is the result of multiplying that number by any whole number.

**What is the method to find the factors of a large number?**

To find the factors of a large number, you can start by dividing it by 2, then by 3, and so on, until you reach a prime number.

**What is the difference between GCF and LCM?**

The GCF is the largest factor that two or more numbers have in common, while the LCM is the least (smallest) multiple that two or more numbers have in common.

**What are the uses of factors and multiples in real life?**

Factors and multiples can be used in many real-life situations, such as finding the number of seats in a theater or the number of tiles needed to cover a floor. They are also important in mathematical concepts like fractions, ratios, and proportions.

**How are factors and multiples related?**

When we multiply two numbers to find the product, each number being multiplied is a factor of the product, whereas the number obtained from the product is a multiple of each number being multiplied.

**What is the definition of factors and multiples?**

Factor is a number that divides another number evenly (with remainder $=$ 0). Multiples are numbers obtained by multiplying a given number by a positive integer.