# Factors and Multiples

## Factors and Multiples

A number's components are all the integers that may divide it in two equally. In other words, a number's factors are the set of all integers that may be used to multiply by the number itself. The 24's factors are as follows:

1, 2, 3, 4, 6, 8, 12, 24 (1 + 24 Equals 2, 2 + 12 = 24, etc.)

It is also possible to list factors as pairs whose products are numbers. The factor pairs for the number 70 are as follows:

1 × 70 2 × 35 5 × 14 7 × 10

## Prime Factors

A prime number is one that can only be divided by itself and by one.

Prime numbers include the following: 2, 3, 5, and 7.

## Prime Factoring

Any non-prime number can be prime factored, or divided into its component prime numbers.

Factoring in primes for 12, 100, and 2048:

12 = 2 × 2 × 3

100 = 2 × 2 × 5 × 5

2048 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2

Finding factors and multiples of numbers, as well as the greatest common factor and least common multiple, are all useful uses of prime factorization. 3) Making fraction operations easier

## GCF (Greatest Common Factor)

In arithmetic tests, the Greatest Common Factor (GCF) is frequently utilized. The biggest integer that is both a factor of both integers is the greatest common factor of two numbers. For instance, the GCF of 12 and 8 is 4, since 3 4 = 12 and 2 4 = 8, respectively, and neither number shares a greater factor.

GCFs can be used to simplify fractions by dividing the numerator and denominator by the GCF of the two values. This will give you the fraction in its simplest form. Here's an illustration:

3654=23 (since 36 and 54 have a GCF of 18)

Factors are simpler than multiples. A number's multiples are all the results of multiplying the original number by any integer. For a given integer, there are only a finite number of factors, but there are an infinite number of multiples. The multiples of 4 include, for instance, 4, 8, 12, 16, 20, and so on.

## LCM (Least Common Multiple)

In arithmetic tests, the Least Common Multiple (LCM) is frequently employed. The smallest integer that is a multiple of both numbers is the least common multiple of the two numbers. The LCM of 12 and 20 equals 60, that of 54 and 9 equals 54, and that of 2 and 3 equals 6.

The LCM is most frequently employed when adding and subtracting fractions since it yields the smallest common denominator when the original denominators are multiplied by each other. The LCM of 8 and 12 in the example below is 24.

18+112=324+224=524

## Finding the Least Common Multiple For a Pair of Numbers

Sometimes, there will be a straight-up least common multiple question: "What is the LCM of 39 and 54?"

Break down each number into its prime factors using a factor tree:

39: 3 × 13

54: 2 × 3 × 3 × 3

Then count quantity of digits within each [digit group] in the factorization of each number:

39: [3] × [13] (one "3" and one "13")

54: [2] × [3 × 3 × 3] (one "2" and three "3")

Now, we have 4 [digit groups]: a single 3, a single 2, a single 13, and three 3's. We can only choose 1 digit group per factor value, and we want to choose the largest digit group for each factor value:

39: [3] × [13]

54: [2] × [3 × 3 × 3]

ONE "2" THREE "3" (grouped under 54, which beats the ONE 3 under 39) ONE "13" Then multiply the winning numbers together:

2 × 3 × 3 × 3 × 13 = 702 = LCM of 39 and 54

Let's try another one:

"What is the LCM of 18 and 20?"

18 = 3 × 3 × 2

20 = 2 × 2 × 5

Winning numbers:

18 = 3 × 3 × 2

20 = 2 × 2 × 5

LCM = 3 × 3 × 2 × 2 × 5 = 180