Chapter Exam

Chapter 2: Numbers and Sequences — Online MCQ Test

Q: According to Euclid's Division Lemma, for any two positive integers a and b, there exist unique integers q and r such that:

Answer: B. a = bq + r, 0 ≤ r < b — Euclid's Division Lemma states that a = bq + r, where q and r are integers and 0 ≤ r < b.

Q: The Fundamental Theorem of Arithmetic states that every composite number can be expressed as:

Answer: A. a product of primes uniquely, apart from the order of the factors — Every composite number has a unique prime factorisation except for the order of the prime factors.

Q: Find the HCF of 135 and 225 using Euclid's algorithm.

Answer: C. 45 — 225 = 135 × 1 + 90, 135 = 90 × 1 + 45, and 90 = 45 × 2 + 0. Therefore, the HCF is 45.

Q: Find the least positive number which leaves remainder 3 when divided by 12, 15 and 20.

Answer: C. 63 — LCM of 12, 15 and 20 is 60. The required number is 60 + 3 = 63.

Q: Find the least positive integer which leaves remainder 5 when divided by 6, 8 and 9, and is also divisible by 7.

Answer: A. 77 — The number is of the form 72k + 5 since LCM(6, 8, 9) = 72. For divisibility by 7, the least value is k = 1, giving 77.

MATHS · CLASS 10th · Tamil Nadu State Board

Practice Chapter 2: Numbers and Sequences with a free chapter-wise online MCQ test for Tamil Nadu State Board CLASS 10th MATHS. This chapter covers: Euclids Division Lemma Fundamental Theorem of Arithmetic Prime factorisation HCF LCM Arithmetic Progression AP Geometric Progression GP Common difference Common ratio nth term Sum.... AI-generated questions from basic to board-exam level, with instant results and explanations.

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Chapter 2: Numbers and Sequences — Important Questions & Answers (FAQ)

Frequently asked questions from Tamil Nadu State Board CLASS 10th MATHS — Chapter 2: Numbers and Sequences, with answers and explanations. These are sample questions; the exam has its own separate question set.

According to Euclid's Division Lemma, for any two positive integers a and b, there exist unique integers q and r such that:

A. a = q + br, 0 < r < b
B. a = bq + r, 0 ≤ r < b ✓
C. b = aq + r, 0 ≤ r < a
D. a = br - q, 0 < r ≤ b

Answer: B. a = bq + r, 0 ≤ r < b
Euclid's Division Lemma states that a = bq + r, where q and r are integers and 0 ≤ r < b.

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as:

A. a product of primes uniquely, apart from the order of the factors ✓
B. a sum of primes uniquely
C. a product of even numbers
D. a difference of two prime numbers

Answer: A. a product of primes uniquely, apart from the order of the factors
Every composite number has a unique prime factorisation except for the order of the prime factors.

Find the HCF of 135 and 225 using Euclid's algorithm.

A. 15
B. 30
C. 45 ✓
D. 90

Answer: C. 45
225 = 135 × 1 + 90, 135 = 90 × 1 + 45, and 90 = 45 × 2 + 0. Therefore, the HCF is 45.

Find the least positive number which leaves remainder 3 when divided by 12, 15 and 20.

A. 33
B. 43
C. 63 ✓
D. 83

Answer: C. 63
LCM of 12, 15 and 20 is 60. The required number is 60 + 3 = 63.

Find the least positive integer which leaves remainder 5 when divided by 6, 8 and 9, and is also divisible by 7.

A. 77 ✓
B. 149
C. 221
D. 293

Answer: A. 77
The number is of the form 72k + 5 since LCM(6, 8, 9) = 72. For divisibility by 7, the least value is k = 1, giving 77.

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