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# Multiples and Factors

Multiples and factors are two important concepts in pre-algebra. Let us look at each of these two concepts in detail below:

## Multiples

Multiples of a number are the numbers into which it can be divided without any remainder. For example 24 ÷ 6 = 4. Here 6 divides into 24 with no remainder.

Multiples are also numbers which are itself or greater than the given number. Multiples of 6 are 6, 12, 18, 24, and so on.

### Least common multiple (LCM)

Least common multiple of two numbers is the smallest number into which both the given numbers can divide. For example the lowest (least) common multiple of 3 and 8 is 24.

To find the least common multiple of two numbers, we follow these steps:

1. List all the multiples of the two given numbers
2. Find the smallest number which is common in both sets.

Let’s try this with an example. What is the LCM of 5 and 9?

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, and so on

Multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, and so on

The smallest number common to both the series of multiples is 45. Hence the LCM of 5 and 9 is 45.

## Factors

The factors of a number are the numbers that divide into the given number without leaving any remainder. 6 is a factor of 18 because 18 ÷ 6 = 3, or 3 x 6 = 18 i.e. 6 divides into 18 without remainder.

### Highest common factor (HCF)

Highest common factor of two (or more) numbers is the number which divides into the given number without a remainder.

To find the highest common factor of two numbers, we follow these steps:

1. List all the factors of the two given numbers
2. Find the largest number which is common in both sets.

Let’s try this with an example. What is the HCF of 18 and 24?

Factors of 18: 18×1, 9×2, 6×3

{18, 9, 6, 3, 2, 1}

Factors of 24: 24×1, 12×2, 8×3, 6×4

{24, 12, 8, 6, 4, 3, 2, 1}

The highest number common to both the series is 6. Hence the HCF of 18 and 24 is 6.

Note: Sometimes HCF is also called Greatest Common Divisor (GCD) and LCM is also called Least Common Multiplier (LCM)