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Class-10 Ch-1 | Real Numbers: Euclid’s Division Lemma

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.

Euclidean division algorithm (is a basis of  Euclid’s division lemma).

To calculate the Highest Common Factor (HCF) of two positive integers a and b we use Euclid’s division algorithm. HCF is the largest number which exactly divides two or more positive integers. That means, on dividing both the integers a and b the remainder is zero.

Euclid’s Division Lemma Algorithm

Example: Consider two numbers 78 and 980 and we need to find the HCF 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 78 and the remainder 44, apply Euclid division lemma again.

78 = 44 × 1 + 34

Similarly, consider the divisor 44 and the remainder 34, apply Euclid division lemma to 44 and 34.

 

44 = 34 × 1 + 10

Following the same procedure again,

34 = 10 × 3 + 4

10 = 4 × 2 + 2

4 = 2 × 2 + 0

We see that the remainder has become zero, therefore, proceeding further is not possible.

Hence, the HCF is the divisor b left in the last step. We can conclude that the HCF of 980 and 78 is 2.

Example:

To find the HCF of two numbers 250 and 75. Here, the larger the integer is 250, therefore, by applying Euclid Division Lemma a = bq + r where 0 ≤ r < b, we have

a = 250 and b = 75

250 = 75 × 3 + 25

By applying the Euclid’s Division Algorithm to 75 and 25, we have:

75 = 25 × 3 + 0

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

What is the HCF of 196 and 38220?

38220 is greater than 196.

Applying Euclid’s division algorithm,
38220 = 196 × 195 + 0
Therefore, the HCF of 196 and 38220 is 196.

What is the HCF of 4052 and 12576?

12576 is greater than 4052.
Applying Euclid’s division algorithm,
12576 = 4052 × 3 + 420
4052 = 420 × 9 + 272
272 = 148 × 1 + 124
148 = 124 × 1 + 24
124 = 24 × 5 + 4
24 = 4 × 6 + 0
Therefore, the HCF of 4052 and 12576 is 4.

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