Question

What is the formula of Euclid Division lemma?????

Answer 1

$\huge\mathfrak\red{Answer}$

• Euclid division lemma :

a = bq + r where q or r can also be zero.

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Answer 2

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Euclid Division lemma

Real Numbers, Class 10 Mathematics. EUCLID'S DIVISION LEMMA OR EUCLID'S DIVISION ALGORITHM. For any two positive integers a and b there exist unique integers q and r such that. a = bq + r, where 0 ≤ r < b

OR

Theorem (Euclid’s Division Lemma):

For a pair of given positive integers ‘a’ and ‘b’, there exist unique integers ‘q’ and ‘r’ such that

a=bq+r, where 0≤r<b

Explanation:

Thus, for any pair of two positive integers a and b; the relation

a=bq+r, where 0≤r<b

will be true where q is some integer.

Example:

(a) 20, 8

Let 20 = a and 8 = b

Therefore, by applying the relation a=bq+ra=bq+r, where 0≤r<b0≤r<b we get 20=8×2+420=8×2+4 (In this q=2q=2 and r=4r=4)

(b) 17, 5

Let 17 = a and 5 = b

Therefore, by applying the relation a=bq+ra=bq+r, where 0≤r<b0≤r<b we get 17=5×3+217=5×3+2 (In this q=3q=3 and r=2r=2

Ex. Question - 1: Use Euclid’s division algorithm to find the HCF of:

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Solution: (i) 135 and 225

In the given problem, let 225 = a, and 135 = b

Therefore, by applying the relation a=bq+ra=bq+r, where 0≤r<b0≤r<b we get

225=135×1+90225=135×1+90 (Here r=90r=90)

Since, r (remainder) is not equal to zero (0). Thus, by applying the Euclid’s division algorithm, by taking 135 = a, and 90 = b we get

135=90×2+45135=90×2+45 (Here r=45r=45)

Since, in this step also, r is not equal to zero(0). Thus by continuing the Euclid’s division algorithm, by taking this time, 90 = a, and 45 = b we get

90=45×2+090=45×2+0 (In this step we get r=0r=0)

Therefore, 45 is the HCF of given pair 225 and 135

Thus, Answer: 45