Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy
a. 1 < r < b
b. 0 < r ≤ b
c. 0 ≤ r < b
d. 0 < r < b

Solution:

Given, a = bq + r

r must satisfy at 0 ≤ r < b

Therefore, by Euclid’s division lemma r must satisfy at 0 ≤ r < b.

✦ Try This: Find the HCF of 81 and 675 using the Euclidean division algorithm

The larger integer is 675, therefore, by applying the Division Lemma a = bq + r where 0 ≤ r < b, we have

a = 675 and b = 81

675 = 81 × 8 + 27.

The divisor is 27.

Therefore, the HCF of 675 and 81 is 27

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 1

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NCERT Exemplar Class 10 Maths Exercise 1.1 Sample Problem 2

Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy, a. 1 < r < b, b. 0 < r ≤ b, c. 0 ≤ r < b, d. 0 < r < b

Summary:

Euclid’s division lemma states that for two positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy 0 ≤ r < b

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☛ Related Questions:

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• For some integer q, every odd integer is of the form a. q, b. q + 1, c. 2q, d. 2q + 1
• n² – 1 is divisible by 8, if n is a. an integer, b. a natural number, c. an odd integer, d. an even . . . .