What is the fastest way to finding the highest common factor (HCF) between two numbers, and how can you check your answer?
The quickest way to find the highest common factor between two numbers is to start by identifying all the factors of the smallest number first, then with this list, starting from the biggest factors, see if each number divides into the larger number without leaving a remainder.
As you progress down through the list of factors (from the smaller number) when you find a number that divides into the larger number you can stop as you have found the highest common factor of both numbers .
For example, with the numbers 42 and 90 the highest common factor is 6.
HCF Of 42 And 90
Here is a step by step guide demonstrating how to get the highest common factor between the numbers 42 and 90:
With the two numbers, identify which is the smallest: 42 or 90?
42 is the smaller number.
With the smaller number 42, find all the factors of this number.
Start by inserting into your factor list both the number 1 and 42.
Working factor list contains 1 and 42
Increment the factor by 1: 1 + 1 = 2
Can 2 be divided into 42 without leaving a remainder?
Yes! Therefore, add both 2 and the quotient 21 into the factor list.
Working factor list contains 1, 2, 21 and 42
Increment the factor by 1: 2 + 1 = 3
Can 3 be divided into 42 without leaving a remainder?
Yes! Therefore, add both 3 and the quotient 14 into the factor list.
Working factor list contains 1, 2, 3, 14, 21 and 42
Increment the factor by 1: 3 + 1 = 4
Can 4 be divided into 42 without leaving a remainder?
No! 4 divided into
Increment the factor by 1: 4 + 1 = 5
Can 5 be divided into 42 without leaving a remainder?
No! 5 divided into
Increment the factor by 1: 5 + 1 = 6
Can 6 be divided into 42 without leaving a remainder?
Yes! Therefore, add both 6 and the quotient 7 into the factor list.
Working factor list contains 1, 2, 3, 6, 7, 14, 21 and 42
As you have reached (or are about to eclipse) the square root of 42 you can stop here.
You now have all the factors of 42 being 1, 2, 3, 6, 7, 14, 21 and 42.
Starting from the end of your factor list, see if each number can be divided into the larger number without any remainder.
Can 42 be divided into 90 without leaving a remainder?
No! 42 divided into 90 leaves a remainder of 6.
Therefore, you need to move to the next factor of the smallest number in the list.
Can 21 be divided into 90 without leaving a remainder?
No! 21 divided into 90 leaves a remainder of 6.
Therefore, you need to move to the next factor of the smallest number in the list.
Can 14 be divided into 90 without leaving a remainder?
No! 14 divided into 90 leaves a remainder of 6.
Therefore, you need to move to the next factor of the smallest number in the list.
Can 7 be divided into 90 without leaving a remainder?
No! 7 divided into 90 leaves a remainder of 6.
Therefore, you need to move to the next factor of the smallest number in the list.
Can 6 be divided into 90 without leaving a remainder?
Yes! 6 can be divided into 90 without leaving a remainder.
Therefore, 6 is the highest common factor of 42 and 90.
As you can see from the example above the step-by-step process in working with the factors of the smallest number can help to quickly achieve the answer of what the highest common factor is for both numbers.
How To Check Your Answer
Is there a way you can check your answer if you’ve used the approach above? Yes, there is!
An alternative approach to finding the highest common factor between two numbers is to find the factors of the largest number first and then see if each factor can be divided into the smallest number. To help quicken the pace in checking your answer you can remove any factors from the larger number that are greater than the smaller number.
Here’s how this approach would work using the same numbers above:
With the two numbers, identify which is the largest of the two numbers: 42 or 90? 90 is the larger of the two numbers.
With this number, find all its factors.
Start by inserting into your factor list both the number 1 and 90.
Working factor list contains 1 and 90
Increment the factor by 1: 1 + 1 = 2
Can 2 be divided into 90 without leaving a remainder?
Yes! Therefore, add both 2 and the quotient 45 into the factor list.
Working factor list contains 1, 2, 45 and 90
Increment the factor by 1: 2 + 1 = 3
Can 3 be divided into 90 without leaving a remainder?
Yes! Therefore, add both 3 and the quotient 30 into the factor list.
Working factor list contains 1, 2, 3, 30, 45 and 90
Increment the factor by 1: 3 + 1 = 4
Can 4 be divided into 90 without leaving a remainder?
No! 4 divided into
Increment the factor by 1: 4 + 1 = 5
Can 5 be divided into 90 without leaving a remainder?
Yes! Therefore, add both 5 and the quotient 18 into the factor list.
Working factor list contains 1, 2, 3, 5, 18, 30, 45 and 90
Increment the factor by 1: 5 + 1 = 6
Can 6 be divided into 90 without leaving a remainder?
Yes! Therefore, add both 6 and the quotient 15 into the factor list.
Working factor list contains 1, 2, 3, 5, 6, 15, 18, 30, 45 and 90
Increment the factor by 1: 6 + 1 = 7
Can 7 be divided into 90 without leaving a remainder?
No! 7 divided into
Increment the factor by 1: 7 + 1 = 8
Can 8 be divided into 90 without leaving a remainder?
No! 8 divided into
Increment the factor by 1: 8 + 1 = 9
Can 9 be divided into 90 without leaving a remainder?
Yes! Therefore, add both 9 and the quotient 10 into the factor list.
Working factor list contains 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90
As you have reached (or are about to eclipse) the square root of 90 you can stop here.
You now have all the factors of 90 being 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45 and 90.
Remove the numbers from this list that are larger than the smaller number 42.
This will mean your factor list now only contains: (1, 2, 3, 5, 6, 9, 10, 15, 18, 30)
Starting from the end of the list, see if each number can be divided into the smaller number without any remainder.
Can 30 be divided into 42 without leaving a remainder?
No! 30 divided into 42 leaves a remainder of 12.
Therefore, you need to move to the next factor in the list.
Can 18 be divided into 42 without leaving a remainder?
No! 18 divided into 42 leaves a remainder of 6.
Therefore, you need to move to the next factor in the list.
Can 15 be divided into 42 without leaving a remainder?
No! 15 divided into 42 leaves a remainder of 12.
Therefore, you need to move to the next factor in the list.
Can 10 be divided into 42 without leaving a remainder?
No! 10 divided into 42 leaves a remainder of 2.
Therefore, you need to move to the next factor in the list.
Can 9 be divided into 42 without leaving a remainder?
No! 9 divided into 42 leaves a remainder of 6.
Therefore, you need to move to the next factor in the list.
Can 6 be divided into 42 without leaving a remainder?
Yes! 6 can be divided into 42 without leaving a remainder.
Therefore, 6 is the highest common factor of 42 and 90.
As you can see both approaches achieve the same answer, which helps to give you confidence with your original answer.
Summary
The highest common factor (HCF) between 2 numbers is the largest number that can be divided into both numbers without leaving a remainder. The HCF of 42 and 90 is 6.
Finding the highest common factor is a skill used quite frequently in arithmetic, especially when reducing fractions.