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The commonly used methods to find the Highest Common Factor (HCF) or the Greatest Common Divisor (GCD) are the prime factorization method and the simultaneous division method. There is another method (less popular) to find HCF which is extremely useful in certain cases but not so useful in the rest.

If the question is, find the HCF of 80 and 216, this is how you can proceed:

Prime factorization:

80 = 2*2*2*2*5
216 = 2*2*2*3*3*3

HCF = 2*2*2 = 8

Simultaneous Division:

After this, as there are no common factors in 10 and 27, we will stop and take 2*2*2 = 8 as HCF.

Division method:

I am illustrating division method step by step.

Step 1: Divide the larger number by the smaller number. So 216 when divided by 80, gives 2 as quotient and 56 as remainder.
Step 2: Divide 80 (divisor of the previous division) by 56. This gives you 1 as quotient, and 24 as remainder.
Step 3: Divide 56 (divisor of the previous division) by 24. This gives you 2 as quotient, and 8 as remainder.
Step 3: Divide 24 (divisor of the previous division) by 8. This gives you 3 as quotient, and 0 remainder. That makes 8 the HCF of 80 and 216.

We will look at another example. Find HCF of 105 and 462.

Step 1: Divide 462 by 105. Quotient is 4 and remainder is 42.
Step 2: Divide 105 by 42. Quotient is 2 and remainder is 21.
Step 3: Divide 42 by 21. Quotient is 2 and remainder is 0. Hence, 21 is the HCF of 105 and 462.

This method is particularly useful when you get numbers that are not easy to factorize. For example, find the HCF of 629 and 851. In the previous examples, we could use the factorization and simultaneous division method as the prime factors were easy to find. But for something like HCF of 629 and 851, this method is super convenient. Give it a try. Answer? 37

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