You were introduced to the world of
real numbers in class 9. In this chapter, we will continue our discussion of real numbers beginning with two important properties of positive integers. One is Euclid's division algorithm which will be dealt with here, another is Fundamental Theorem of Arithmetic which will be dealt with in the next lesson.
A dividend can be written as, Dividend = Divisor × Quotient + Remainder. This brings to Euclid's division lemma.
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:
- 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.
- 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’.
- 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.