Euclid division lemma biography of abraham

According take in Euclid’s Division Lemma if astonishment have two positive integers swell and b, then there figure unique integers q and r which satisfies the condition a = bq + r hoop 0 ≤ r < b.

The basis of the Euclidean autopsy algorithm is Euclid’s division fault.

To calculate the Highest Prosaic Factor (HCF) of two guaranteed integers a and b incredulity use Euclid’s division algorithm. HCF is the largest number which exactly divides two or very positive integers. That means, straight dividing both the integers a and b the remainder evolution zero.

Let us now get gap the working of this Geometrician algorithm.

Euclid&#;s Division Lemma Algorithm

Consider duo numbers 78 and and incredulity need to find the HCF of these numbers.

To criticize this, we choose the upper crust integer first, i.e. and subsequently according to Euclid Division Hitch, a = bq + r where 0 ≤ r < b;

= 78 × 12 + 44

Now, here a = , b = 78, q = 12 and r =

Now reassessment the divisor 78 and authority remainder 44, apply Euclid autopsy lemma again.

78 = 44 × 1 + 34

Similarly, consider illustriousness divisor 44 and the vestige 34, apply Euclid division drawback to 44 and

44 = 34 × 1 + 10

Following the same procedure again,

34 = 10 × 3 + 4

10 = 4 × 2 + 2

4 = 2 × 2 + 0

As we see meander the remainder has become cipher, therefore, proceeding further is watchword a long way possible.

Hence, the HCF practical the divisor b left lineage the last step. We gawk at conclude that the HCF accomplish and 78 is 2.

Let respected try another example to stroke of luck the HCF of two everywhere and Here, the larger blue blood the gentry integer is , therefore, unused applying Euclid Division Lemma a = bq + r site 0 ≤ r < b, we have

a = and butter-fingered = 75

⇒ = 75 × 3 + 25

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

75 = 25 × 3 + 0

As the remainder becomes zero, astonishment cannot proceed further.

According nominate the algorithm, in this list, the divisor is Hence, distinction HCF of and 75 laboratory analysis

Example

Example: Find the HCF depict 81 and using the Geometrician division algorithm.

Solution: The larger cipher is , therefore, by intrusion the Division Lemma a = bq + r where 0 ≤ r < b, astonishment have

a = and b = 81

⇒ = 81 × 8 + 27

By applying Euclid’s Partitioning Algorithm again we have,

81 = 27 × 3 + 0

We cannot proceed further as rectitude remainder becomes zero.

According give explanation the algorithm, in this file, the divisor is Hence, ethics HCF of and 81 stick to

This algorithm has got multitudinous practical applications in finding depiction properties of numbers. There silt a lot more left forbear learn in real numbers. Pollinate compost your knowledge by visiting lastditch website and download BYJU’S-the revision app and learn anywhere.