Euclid’s Division Lemma

 

Euclid’s division lemma, states that for any two positive integers ‘a’ and ‘b’ we can find two whole numbers ‘q’ and ‘r’ such that a = b × q + r  where 0 ≤ r < b.
Euclid’s division lemma can be used to find the highest common factor of any two positive integers and to show the common properties of numbers.

The following steps to obtain H.C.F using Euclid’s division lemma:

1. Consider two positive integers ‘a’ and ‘b’ such that a > b. Apply Euclid’s division lemma to the given integers ‘a’ and ‘b’ to find two whole numbers ‘q’ and ‘r’ such that, a = b x q + r.

2. Check the value of ‘r’. If r = 0 then ‘b’ is the HCF of the given numbers. If r ≠ 0, apply Euclid’s division lemma to find the new divisor ‘b’ and remainder ‘r’.

3. Continue this process till the remainder becomes zero. In that case the value of the divisor ‘b’ is the HCF (a , b). Also HCF(a ,b) = HCF(b, r).

Euclid’s division algorithm can also be used to find some common properties of numbers.

Example:

(a) 20, 8

Let 20 = a and 8 = b

Therefore by applying the relation a = b X q + r,

Where 0 < r <b we get

20 = 8 X 2+4 ( in this q = 2 and r = 4)

(b) 17, 5

Let 17 = a and 5 = b

Therefore by applying the relation a = b X q + r,

Where 0 < r <b we get

17 = 5  X 3 +2 ( in this q = 3 and r = 2)