Class 10 Euclid's Division Lemma — practice questions with answers
Number Systems · 5 sample questions from the school syllabus. Try each one, then open the answer and the worked solution.
1. Euclid's Division Lemma states that for positive integers a and b, there exist unique integers q and r such that a = bq + r where:
A0 <= r < b
B0 < r <= b
C0 <= r <= b
D1 <= r < b
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Answer: A) 0 <= r < b
By Euclid's Division Lemma, the remainder r always satisfies 0 <= r < b, where b is the divisor.
2. When a = 0 and b = 7 in Euclid's Division Lemma a = bq + r, the values of q and r are:
Aq = 0, r = 0
Bq = 1, r = 0
Cq = 0, r = 7
Dq = 7, r = 0
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Answer: A) q = 0, r = 0
0 = 7 x 0 + 0, so q = 0 and r = 0.
3. Find the HCF of 441 and 567 using Euclid's algorithm.
A57
B63
C69
D126
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Answer: B) 63
567 = 441 x 1 + 126; 441 = 126 x 3 + 63; 126 = 63 x 2 + 0. HCF = 63.
4. Using Euclid's Division Lemma, any positive integer is 5q, 5q+1, 5q+2, 5q+3 or 5q+4. For how many of these five forms is the square of the integer of the form 5m (a multiple of 5)?
A1
B3
C0
D2
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Answer: A) 1
Squares: (5q)² = 5m; others give remainders 1,4,4,1 mod 5. Only 5q gives a square divisible by 5. So 1 form.
5. Find the largest number which divides 626, 3127 and 15628 leaving remainders 1, 2 and 3 respectively. Use Euclid's algorithm after subtracting the remainders.
A625
B565
C685
D1250
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Answer: A) 625
Subtract remainders: 626 - 1 = 625, 3127 - 2 = 3125, 15628 - 3 = 15625. HCF(625, 3125) = 625 since 3125 = 625 x 5. HCF(625, 15625) = 625 since 15625 = 625 x 25. So the required number is 625.