Question

get 20 points | real numbers euclid division lemma notes​

Answer 1

Euclid's Division Lemma states that, if two positive integers “a” and “b”, then there exists unique integers “q” and “r” such that which satisfies the condition a = bq + r where 0 ≤ r ≤ b. ... In this example, 9 is the divisor, 58 is the dividend, 6 is the quotient and 4 is the remainder.

Answer 2

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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.

following steps to obtain HCF of two numbers by 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).