Prime factorization of 30 and 90
Prime factorization of 30 and 90

Factors of 30 are those integers that divide 30 without leaving any remainder. The factors of 30 cannot be a decimal or a fraction.

For example, 2 is a factor of 30 because when we divide 30 by 2 we get the remainder as 0. The quotient will be 15, which is also a factor of 30.

Factors

Factor pairs

Prime factorization

1, 2, 3, 5, 6, 10, 15, 30

(1, 30), (2, 15), (3, 10)

and (5,6)

30 = 2 \(\times \) 3 \(\times \) 5

We can find the factors of 30 by using divisibility rules and division facts. The factors can be obtained by the following method.

Divisor

Is the number a factor of 30 ?

Multiplication Equation

1

Yes, 1 is a factor of every number

1 × 30 = 30

2

Yes, 30 is even

2 × 15 = 30

3

Yes, 3 + 0 = 3 is divisible by 3

3 × 10 = 30

4

No, 30 ÷ 4 = 7 R2

5

Yes, 30 ÷ 5 = 6 R0

5 × 6 = 30

6

Yes, 30 is even and divisible by 3

6 × 5 = 30

The factor tree below is used to derive the prime factorization of 30:

From the factor tree we can see that the prime factorization of 30 is 2 × 3 × 5.

Therefore, 2, 3 and 5 are the prime factors of 30.

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Factor pairs of 30 are pairs of factors of 30 which when multiplied together result in 30.

Factors of 30: 1, 2, 3, 5, 6, 10, 15 and 30.

Positive factors of 30

Positive factor pairs of 30

1 \(\times \) 30

(1, 30)

2 \(\times \) 15

(2, 15)

3 \(\times \) 10

(3, 10)

5 \(\times \) 6

(5, 6)

Example 1: Find the common factors of 20 and 30.

Solution:

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Factors of 20: 1, 2, 4, 5, 10, 20.

So, the common factors of 20 and 30 are 1, 2, 5 and 10. So 20 and 30 have 4 common factors.

Example 2: How many factors do 30 have?

Solution:

Factors of 30: 1, 2, 3, 5, 6, 10, 15 and 30.

So, there are 8 factors for 30.

Example 3: There are 30 fish in an aquarium. Joann checks the colors of the fish and notices that there are fishes of 6 different colors. It is known that fishes can be divided into groups depending on their colors such that each group has an equal number of fishes, how many fishes are there in each color group?

Solution:

Number of fishes = 30

Number of colors = 6

6 is a factor of 30. Hence, we can divide 30 fishes equally into 6 color groups.

Number of fish in each group of color = \(\frac{30}{6} \) = 5 R0

Therefore, 5 fish will be in each color group.

No, 9 is not a factor of 30. As the number 9 does not divide 30 exactly, it leaves a remainder of 3. Hence, 9 is not a factor of 30.

Yes, 30 is a composite number as it has factors other than one and itself. It has factors of 2, 3, 5, 6, 10 and 15 other than 1 and 30.

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

Hence, the sum of the all the factors of 30 is = 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72

Hence, the sum is 72.

Factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30 and factors of 18 are 1, 2, 3, 6, 9, and 18.

So, the common factors of 30 and 18 are 1, 2, 3 and 6.

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