Euclid’s Division Lemma:

According to Euclid’s Division Lemma if we have two positive integers a and b, then there exists unique integersqandrwhich satisfies the conditiona = bq + rwhere 0≤ r ≤ b.

The basis of Euclidean division algorithm is Euclid’s division lemma. To calculate the Highest Common Factor (HCF) of two positive integersaandbwe use Euclid’s division algorithm. HCF is the largest number which exactly divides two or more positive integers. By exactly we mean that on dividing both the integersaandbthe remainder is zero.

Let us now get into the working of this euclidian algorithm.

Consider we have two numbers 78 and 980 and we need to find the HCF of both of these numbers. To do this, we choose the largest integer first, i.e. 980 and then according to Euclid DivisionLemma,a = bq + rwhere 0 ≤r≤b;

980 = 78 × 12 + 44

Now, herea= 980,b= 78,q= 12 andr= 44.

Now consider the divisor as 78 and the remainder 44 and apply the Euclid division method again, we get

78 = 44 × 1 + 34

Similarly, consider the divisor as 44 and the remainder 34 and apply the Euclid division method again, we get

44 = 34 × 1 + 10

Following the same procedure again,

34 = 10 × 3 + 4

10=4×2+2

4=2×2+0

As we see that the remainder has become zero therefore proceeding further is not possible and hence the HCF is the divisorbleft in the last step which in this case is 2. We can say that the HCF of 980 and 78 is 2.