Here we are going to discuss, what are the **factors of 68** and how many factors 68 has and what are the **factor pairs of 68**. Here are some methods, showing how to find the factors of 68.

*Actually, Factor is a number which divides any number completely without leaving remainder.*

*Or simply we can say that if we multiply two whole numbers (positive & negative) and it gives a product. Then **the numbers that we multiply, are the factors of the product.*

For example, if we multiply 4 to 17, it gives 68. Here 4 and 17 are positive factors of 68 or we multiply -2 to -34, it gives 68. So -2 and -34 are negative factors of 68.

Table of Contents

## What are the Factors of 68 ?

Since factors of 68 are the numbers, which divides 68 completely without leaving remainder. So here we are

going to divide 68 by all numbers up to 68. Let’s see the process –

We have to stop the process, if any number of column ‘B’ appears in column ‘A’.

Here we stop the division when 17 appears in column ‘A’, which has already been came in column ‘B’.

Take another example, Factors of 25

In above example, no element of Column ‘B’ appears in Column ‘A’, but we stopped the process

when 5 comes, because on dividing by 5, it produces the same number (5 = 5) .

So we have to stop the process when any element comes in both columns at same level.

And this element will consider as a factor.

*Note :- This situation occurs with a perfect square number, such as 9, 25, 36 etc.*

Hence, all the factors of 68 in above table are : 1, 2, 4, 17, 34, 68

Similarly, negative factors of 62 are : -1, -2, -4, -17, -34, -68

## Factor pairs of 68

As we saw in above table, 1 multiply by 68 gives 68. So (1, 68) or (68, 1) is a factor pair of 68.

Similarly, (2, 34) and (4, 17) are another factor pairs of 68.

In example of 25 ; (1, 25) or (25, 1) and (5, 5) are factor pairs.

## Methods of Prime Factors of 68

Here we will discuss, mainly two methods of factorization to find Prime Factors of any number.

### Method #1 (Division Method)

**Step 1 :** Divide the number by the smallest prime number, which divides the number exactly.

**Step 2 :** Divide the quotient again by the smallest prime number. If quotient is not exactly divisible by the

smallest prime number, choose next smallest prime number.

**Step 3 :** Repeat Step 2 again and again till the quotient becomes 1.

**Step 4 :** All the prime numbers used in above process are the prime factors.

Check : Multiply all the prime factors, the multiplication should be the number itself.

Caution : In this method, only prime numbers will be use to divide.

Take example of 68 and 180

In above example of 68,

Step 1 : First of all, we divide 68 by smallest prime number i.e. 2, it gives 34 as quotient and remainder is zero.

(Note – if remainder will not zero, means doesn’t divide exactly, try next prime number to divide)

Step 2: Again we have to divide the quotient 34 by a prime number i.e. 2. It gives 17 as quotient and remainder is zero.

Step 3: Again we have to divide the quotient 17 by a prime number. Since 17 is a prime number, hence 17 will divide by itself only. So we divide 17 by 17.

Step 4. We got 1 as a quotient.

Hence, 68 = 2 x 2 x 17. So **68 has only two prime factor i.e. 2 and 17.**

In second example of 180

Step 1 : First of all, we divide 180 by smallest prime number i.e. 2 and we got 90 as quotient and remainder is zero. (Note – if remainder will not zero, means doesn’t divide exactly, try next prime number to divide)

Step 2: Again we divide the quotient 90 by smallest prime number i.e. 2 and we got 45 as quotient.

Again we divide the quotient 45 by smallest prime number i.e. 2, but 2 doesn’t divide 45 exactly. So we try to divide next prime number i.e. 3

Step 3: Repeat step 2 until we got 1 as a quotient.

Hence, 180 = 2 x 2 x 3 x 3 x 5 = 2^{2} x 3^{2}^{ }x 5

So 180 has three prime factor i.e. 2, 3 & 5

### Method #2 (Factor Tree Method)

**Step 1 :** Write a number as root of the tree

**Step 2 :** Find the factors of the number, each factor will be considered as root again (now, we got 2 new root)

**Step 3 :** Repeat step 2 until we can’t factor any more (or until we got prime factor)

**Step 4 :** The end nodes are the prime factors.

**Factor Tree of 68**

We can initiate the tree by choosing any two factors. Here we choose 4 and 17 as factors of 68.

Further, we have to divide these nodes ( 4 & 17). So we split 4 into two factors 2 and 2.

Here we got, 2 and 17 as unique nodes. So 68 has two prime factors i.e. 2 and 17.

*Note:- The Factor Tree may be made so many types depends upon choosing the initial and middle factors.*

## Number of factors of 68

To find the number of factors, we use **Method #1 (Division Method).**

In our example, as we saw above 68 = 2 x 2 x 17 means 2 to the power 2 multiply 17 to the power 1

** Number of factors is = (power + 1)(power + 1)(power + 1)……….** = (2 + 1)(1 + 1) = 3 x 2 = 6

if we consider both positive and negative factors, then

total no. of factors = 6 (positive) + 6 (negative) = 12

Take another example, 180 = 2^{2} x 3^{2}^{ }x 5

Number of factors of 180 is = (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 =18

and if we consider both positive and negative factors, then

total no. of factors = 18 (positive) + 18 (negative) = 36

I hope, this step-by-step tutorial will be helpful to understand factorization method of 68.

Try another number 62 for factors.

Take a new look of factorization on Quora.

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