Maths Olympiad - Number System (Class 9 Mathematics)

IMO and NSO-style challenging problems on number theory, rational and irrational numbers, and algebraic properties. Advanced logical reasoning required.

Maths Olympiad - Number System (Class 9 Mathematics)

IMO and NSO-style challenging problems on number theory, rational and irrational numbers, and algebraic properties. Advanced logical reasoning required.

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TL;DR: IMO and NSO-style challenging problems on number theory, rational and irrational numbers, and algebraic properties. Advanced logical reasoning require…

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IMO and NSO-style challenging problems on number theory, rational and irrational numbers, and algebraic properties. Advanced logical reasoning required.

Maths Olympiad - Number System MCQs with Answers & Explanations

Q1. What is the largest prime number less than 100 that when divided by 7 leaves a remainder of 3?

  1. 83
  2. 79
  3. 73 ✓ (correct)
  4. 97

Explanation: Check: 73/7 = 10 R 3 (73 = 10*7 + 3). Is 73 prime? Yes. 79/7 = 11 R 2, not 3. 83/7 = 11 R 6, not 3. 97/7 = 13 R 6, not 3. So 73 is correct.

Q2. What is the remainder when 2^100 is divided by 5?

  1. 2
  2. 4
  3. 0
  4. 1 ✓ (correct)

Explanation: Finding pattern: 2^1 mod 5 = 2, 2^2 mod 5 = 4, 2^3 mod 5 = 3, 2^4 mod 5 = 1, 2^5 mod 5 = 2. Pattern repeats with cycle 4. 100 mod 4 = 0, so 2^100 mod 5 = 2^4 mod 5 = 1.

Q3. Express 0.3333... (repeating) as a fraction in lowest terms.

  1. 1/3 ✓ (correct)
  2. 2/3
  3. 1/2
  4. 3/10

Explanation: Let x = 0.3333. Then 10x = 3.3333. Subtracting: 9x = 3, so x = 3/9 = 1/3.

Q4. Which of these is a rational number?

  1. sqrt(3)
  2. 0.256256256... ✓ (correct)
  3. pi
  4. sqrt(2)

Explanation: 0.256256256... is a repeating decimal, which is rational. It equals 256/999. sqrt(2), pi, and sqrt(3) are irrational.

Q5. If 2^a * 3^b * 5^c = 360, what is a + b + c?

  1. 5
  2. 3
  3. 6 ✓ (correct)
  4. 4

Explanation: Factorising: 360 = 2^3 x 3^2 x 5^1, so a = 3, b = 2, c = 1 and a + b + c = 6.

Q6. What is the GCD of 84 and 126?

  1. 6
  2. 21
  3. 12
  4. 42 ✓ (correct)

Explanation: Using Euclidean algorithm: 126 = 84*1 + 42, 84 = 42*2 + 0. GCD = 42.

Q7. The LCM of two numbers is 60 and their GCD is 5. If one number is 20, what is the other number?

  1. 15 ✓ (correct)
  2. 30
  3. 12
  4. 25

Explanation: LCM x GCD = product of the numbers: 60 x 5 = 300. The other number is 300 / 20 = 15.

Frequently Asked Questions

How do we determine if a number is rational or irrational?

A rational number can be expressed as a fraction p/q where p and q are integers and q is not zero. It either terminates (like 0.5) or repeats as a decimal (like 0.333...). An irrational number cannot be expressed as a simple fraction and its decimal representation never terminates or repeats (like pi = 3.14159...). Examples of irrational numbers include sqrt(2), sqrt(3), pi, and e.

What is the Fundamental Theorem of Arithmetic?

Every integer greater than 1 can be uniquely expressed as a product of prime numbers (ignoring the order of factors). For example, 12 = 2^2 * 3, and no other prime factorization equals 12. This theorem guarantees that prime factorization is unique, making it a fundamental tool in number theory for finding GCD, LCM, and solving divisibility problems.

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