Factors of 169 Explained with Factor Pairs and Proof

Natural numbers that can divide the number 169 into equal parts are known as its factors. The product of two natural numbers is 169, then the 2 numbers are the required factors. 169 is the perfect square of 13, since, $13\times 13=169$ and 13 is an odd prime number. Also, 169 is a composite number. The pair factors of the number 169 can be expressed in positive form or negative form. For example, the pair factors of the number 169 are represented as $\left ( 1,169 \right )$ or $\left ( -1,-169 \right )$. We multiply the pair of negative numbers, such as multiplying $-1$ and $-169$, and we will get the original number.

Finding the Factors of 169

A factor of 169 is another integer that can divide the integer without leaving any remainder i.e. zero remainder. Therefore, the way you find and list all of the factors of 169 is to go through every number up to and including 169 and check which numbers result in an even quotient (that means no decimal place).

Below are all of the factors of 169:

$169 \div 1 = 169$

$169 \div 13 = 13$

$169 \div 169 = 1$

All of these factors can be used to divide 169 by and get a whole number. The full list of positive factors for 169 are $1, 13$, and $169.$

Prime Factors of 169 : Prime Factorisation

Prime factorisation or prime factor decomposition is the process of finding which prime numbers can be multiplied together to get the original number.

Factor Tree of 169

The prime factors are: $13 \times 13$ or also written as $\left \{ 13,13 \right \}$

Written in exponential form: $13^{2}$

Pair Factors of 169

Two numbers, when multiplied together, equal 169 make for the pair factors of the number. The list of all the 169 factor pairs is provided below. We can see that when we multiply all the factor pairs together, we get 169.

$1 \times 169 = 169$

$13 \times 13 = 169$

$169 \times 1 = 169$

All Factors of 169

The factors of 169 are numbers that when divided by 169, leave zero as remainders. Those that can divide 169 exactly i.e. factors are $1, 13, 169$.

Solved Examples

Example 1: There are 169 fruits to be divided among 13 children. Using the factors of 169, calculate the number of fruits each child gets.

Solution: From the factors of the number 169, we know that $169 = 1 \times 13 \times 13$

$\therefore 169 = 13 \times 13$

Therefore, when there are $13$ children, each child will get $13$ fruits.

Example 2: Find the highest common factors of 13 and 169.

Solution: The prime factorisation of $13$ and 169 is given by

$13 = 1 \times 13$

$169 = 1 \times 13 \times 13$

To find the HCF, we multiply the factors which are common to both $13$ and 169.

$HCF\left ( 13,169 \right )=1 \times 13 = 13$

Practise Questions

1. What is the LCM of 169 and $196$?

1. 33153

2. 33124

3. 22341

4. 32431

Ans: B

2. What is the square root of 169?

1. 13

2. 14

3. 12

4. 16

Ans: A

Conclusion

In this article, we learnt that the factors of 169 are numbers that, when divided by 169, leave zero as remainders. The number 169 has a total of 3 factors: $1, 13, 169$. Its prime factors are $13 \times 13$. The factors in pair for 169 are $\left ( 1,169 \right )$, $\left ( 13,13 \right )$ and $\left ( 169,1 \right )$. You will now be able to factor 169, we hope this article was helpful.