Real numbers - Example 1

Example 1: Use Euclid’s algorithm to find the HCF of 4052 and 12576.
Solution :

Step 1 : Since 12576 > 4052, we apply the division lemma to 12576 and 4052, to get
12576 + 4052 x 3 + 420
Step 2 : Since the remainder 420 0, we apply the division lemma to 4052 and 420, to get
4052 = 420 x9 + 272
Step 3 : We consider the new divisor 420 and the new remainder 272, and apply the division lemma to get
420 = 272 x 1 + 148
We consider the new divisor 272 and the new remainder 148, and apply the division lemma to get
272 = 148 x 1 + 124
We consider the new divisor 148 and the new remainder 124, and apply the division lemma to get
148 = 124 x 1 + 24
We consider the new divisor 124 and the new remainder 24, and apply the division lemma to get
124 = 24 x 5 + 4
We consider the new divisor 24 and the new remainder 4, and apply the division lemma to get
24 = 4 x6 +0
The remainder has now become zero,, so our procedure stops. Since the divisor at this stage is 4, the HCF of 12576 and 4052 is 4.
Notice that 4 = HCF (24,4) = HCF (124,24) = HCF (148, 124) = HCF (272 ,148) = HCF (420, 272) = HCF (4052, 420) = HCF (12576 , 4052).
Euclid’s division algorithem is not only useful for calculating the HCF of very large numbers, but also because it is one of the earliest examples of an algorithm that a computer has been programmed to carry out.

Remarks :
1. Euclid’s division lemma and algorithm are so closely interlinked that people often call former as the division algorithm also.
2. Although Euclid’s Division Algorithm is stated for only positive integers, it can be extended for all integers except zero i.e, b0. However , we shall not discuss this aspect here.