Factors of 210

The factors of 210 are the numbers that divide 210 completely without leaving a remainder. In other words, the numbers that are multiplied together in pairs, resulting in 210 are the factors of 210. The factors or the pair factors of 210 can be positive or negative, but cannot be in fraction or decimal form. For example, the pair factor of 210 is expressed as (1, 210) or (-1, -210). The multiplication of a pair of negative numbers, such as multiplying -1 and -210, results in the original number 210. Here, we are going to discuss the factors of 210, pair factors and the prime factors of 210 using the prime factorization method and many solved examples.

Table of Contents:

What are the Factors of 210?

The numbers that divide 210 evenly and leave a remainder 0 are the factors of 210. As 210 is an even composite number, it has many factors other than 1 and 210. Hence, the factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210. Likewise, the negative factors of 210 are -1, -2, -3, -5, -6, -7, -10, -14, -15, -21, -30, -35, -42, -70, -105 and -210.

Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

Prime Factorization of 210: 2 × 3 × 5 × 7.

Pair Factors of 210

As 210 is a composite number, it has many pair factors. A pair factor of 210 is a pair of numbers, which are multiplied together resulting in the number 210. The positive and negative pair factors of 210 are given below:

Positive Pair Factors of 210:

Positive Factors of 210

Positive Pair Factors of 210

1 × 210

(1, 210)

2 × 105

(2, 105)

3 × 70

(3, 70)

5 × 42

(5, 42)

6 × 35

(6, 35)

7 × 30

(7, 30)

10 × 21

(10, 21)

14 × 15

(14, 15)

Therefore, the positive pair factors of 210 are (1, 210), (2, 105), (3, 70), (5, 42), (6, 35), (7, 30), (10, 21) and (14, 15).

Negative Pair Factors of 210:

Negative Factors of 210

Negative Pair Factors of 210

-1 × -210

(-1, -210)

-2 × -105

(-2, -105)

-3 × -70

(-3, -70)

-5 × -42

(-5, -42)

-6 × -35

(-6, -35)

-7 × -30

(-7, -30)

-10 × -21

(-10, -21)

-14 × -15

(-14, -15)

Hence, the negative pair factors of 210 are (-1, -210), (-2, -105), (-3, -70), (-5, -42), (-6, -35), (-7, -30), (-10, -21) and (-14, -15).

Factors of 210 by Division Method

The factors of 210 using the division method are found by dividing the number 210 by different integers. If the integers divide 210 exactly without leaving a remainder, then those integers are the factors of 210. Now, let us discuss how to find the factors of 210 using the division method.

  • 210/1 = 210 (Factor is 1 and Remainder is 0)
  • 210/2 = 105 (Factor is 2 and Remainder is 0)
  • 210/3 = 70 (Factor is 3 and Remainder is 0)
  • 210/5 = 42 (Factor is 5 and Remainder is 0)
  • 210/6 = 35 (Factor is 6 and Remainder is 0)
  • 210/7 = 30 (Factor is 7 and Remainder is 0)
  • 210/10 = 21 (Factor is 10 and Remainder is 0)
  • 210/14 = 15 (Factor is 14 and Remainder is 0)
  • 210/15 = 14 (Factor is 15 and Remainder is 0)
  • 210/21 = 10 (Factor is 21 and Remainder is 0)
  • 210/30 = 7 (Factor is 30 and Remainder is 0)
  • 210/35 = 6 (Factor is 35 and Remainder is 0)
  • 210/42 = 5 (Factor is 42 and Remainder is 0)
  • 210/70 = 3 (Factor is 70 and Remainder is 0)
  • 210/105 = 2 (Factor is 105 and Remainder is 0)
  • 210/210 = 1 (Factor is 210 and Remainder is 0)

If we divide 210 by any numbers other than 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210, it leaves a remainder of some value. Hence, the factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

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Prime Factorization of 210

The process of writing the number 210 as a product of its prime factors is called the prime factorization of 210. The prime factors of 210 are found as follows:

Take a pair factor of 210, say (1, 210).

We know that the number 1 cannot be factored further. So take the other factor, 210, which is a composite number. Now, find two numbers, which on multiplication results in 210.

Thus, 210 is written as the product of 14 and 15.

Here, both the number 14 and 15 are composite numbers, which can be split further.

Thus, 14 is written as 2× 7, and 15 is written as 3 × 5, where 2, 3, 5 and 7 are prime numbers.

Now, write the number 210 as the product of its prime factors.

Therefore, the prime factorization of 210 is 2 × 3 × 5 × 7.

Solved Examples

Example 1:

Find the common factors of 210 and 120.

Solution:

The factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120.

Hence, the common factors of 210 and 120 are 1, 2, 3, 5, 6, 10, 15 and 30.

Example 2:

Find the common factors of 210 and 209.

Solution:

Factors of 210 = 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

Factors of 209 = 1, 11, 19 and 209.

Thus, the common factor of 210 and 209 is 1.

Example 3:

Find the common factors of 210 and 211.

Solution:

The factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

The factors of 211 are 1 and 211.

As the number 211 is a prime number, the common factor of 210 and 211 is 1.

Stay tuned with BYJU’S – The Learning App to learn more about factors of different numbers and other important mathematical concepts with ease.


Frequently Asked Questions on Factors of 210

What are the factors of 210?

The factors of 210 are 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105 and 210.

What is the prime factorization of 210?

The prime factorization of 210 is 2 × 3 × 5 × 7.

What are the positive pair factors of 210?

The positive pair factors of 210 are (1, 210), (2, 105), (3, 70), (5, 42), (6, 35), (7, 30), (10, 21) and (14, 15).

What are the negative pair factors of 210?

The negative pair factors of 210 are (-1, -210), (-2, -105), (-3, -70), (-5, -42), (-6, -35), (-7, -30), (-10, -21) and (-14, -15).

Is 70 a factor of 210?

Yes, 70 is a factor of 210. If 210 is divisible by 70, it leaves a quotient of 3 and remainder 0, and hence, 70 is a factor of 210.

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