Question

explain euclids division lemma in simple way​

Answer 1

Answer:

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Answer 2

Answer:

According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r ≤ b.

Euclid’s Division Lemma 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 Division Lemma, a = bq + r where 0 ≤ r ≤ b;

980 = 78 × 12 + 44

Now, here a = 980, b = 78, q = 12 and r = 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 divisor b left in the last step which in this case is 2. We can say that the HCF of 980 and 78 is 2.

Let us try another example to find the HCF of two numbers 250 and 75. As the larger the integer is 250, therefore, applying Euclid Division Lemma a = bq + r where 0 ≤ r ≤ b, we have

a = 250 and b = 75

⇒ 250 = 75 × 3 + 25

Applying the Euclid’s Division Algorithm again we have,

75 = 25 × 3 + 0

As the remainder becomes zero, we cannot proceed further. According to the algorithm, in this case the divisor is 25 and hence, the HCF of 250 and 75 is 25.

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