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About Number Theory — Class 10 IB

Explore prime numbers, divisibility, modular arithmetic, and their applications. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 1). On this page you can practice 40 questions across three difficulty levels — 10 easy, 10 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Number Theory — solved examples for Class 10 IB

Example 1easy

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes. Which aspect of this theorem is most crucial for solving problems involving HCF (Highest Common Factor) and LCM (Least Common Multiple) effectively?
  1. A)That composite numbers can be broken down into factors.
  2. B)That only prime numbers are used in the factorization.
  3. C)That the prime factorization is unique, regardless of the order of the prime factors.
  4. D)That prime numbers are infinite.

Step-by-step solution

  1. The uniqueness of prime factorization ensures that every composite number has exactly one set of prime factors and their corresponding powers.
  2. This property is fundamental because it allows us to compare and combine prime factorizations consistently to find HCF (by taking common prime factors with minimum powers) and LCM (by taking all prime factors with maximum powers).
  3. Without this uniqueness, HCF and LCM calculations would not be consistently defined or reliable.

Answer: That the prime factorization is unique, regardless of the order of the prime factors.

Example 2medium

According to Euclid's Division Lemma, for any positive integer 'a' and 'b = 3', which of the following forms correctly represents 'a'?
  1. A)3q, 3q+1, 3q+2
  2. B)3q, 3q+1, 3q+2, 3q+3
  3. C)3q, 3q+1
  4. D)3q+1, 3q+2

Step-by-step solution

  1. Euclid's Division Lemma states that for any two positive integers 'a' and 'b', there exist unique integers 'q' (quotient) and 'r' (remainder) such that a = bq + r.
  2. The crucial condition for the remainder 'r' is 0 ≤ r < b.
  3. Given b = 3, the possible values for the remainder 'r' are 0, 1, or 2.
  4. Therefore, any positive integer 'a' can be represented in the forms 3q + 0 (i.e., 3q), 3q + 1, or 3q + 2.

Answer: 3q, 3q+1, 3q+2

Example 3hard

Consider a positive integer N. If N has an odd number of distinct positive factors, which of the following must be true about N?
  1. A)N is a prime number.
  2. B)N is a perfect square.
  3. C)N is a power of 2.
  4. D)N is a multiple of 3.

Step-by-step solution

  1. Let the prime factorization of N be p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ.
  2. The number of distinct positive factors of N is given by the product (a₁+1)(a₂+1)...(aₖ+1).
  3. For this product to be an odd number, each factor (aᵢ+1) must be odd. If (aᵢ+1) is odd, then each exponent aᵢ must be an even number.
  4. If all exponents aᵢ in the prime factorization are even, it means that N is a perfect square (e.g., N = (p₁^(a₁/2) × p₂^(a₂/2) × ... × pₖ^(aₖ/2))²).

Answer: N is a perfect square.

Practice questions on Number Theory

  1. Q1.easy

    Rohan wanted to check if a number N is divisible by 12. He reasoned that since 12 = 3 × 4, he only needs to check if N is divisible by 3 and by 4. Is Rohan's reasoning correct? If not, why?
    1. A)No, his reasoning is incorrect because he should check divisibility by 2 and 6 instead.
    2. B)Yes, his reasoning is correct because 3 and 4 are coprime factors of 12.
    3. C)Yes, his reasoning is correct because 3 and 4 are factors of 12.
    4. D)No, his reasoning is incorrect because he needs to check divisibility by 12 directly, not by its factors.
    Show answer

    Answer: Yes, his reasoning is correct because 3 and 4 are coprime factors of 12.

    Hint: Consider the relationship between the factors chosen and their common factors. What special property must these factors have?

  2. Q2.easy

    The HCF of two numbers is 8 and their LCM is 144. If one of the numbers is 32, what is the other number?
    1. A)16
    2. B)24
    3. C)48
    4. D)36
    Show answer

    Answer: 36

    Hint: Recall the fundamental relationship between the HCF, LCM, and the product of two numbers.

  3. Q3.easy

    When a positive integer 'n' is divided by 5, the remainder is 2. Which of the following statements correctly expresses this relationship?
    1. A)n ≡ 2 (mod 5)
    2. B)n = 5k + 2, where k is an integer
    3. C)5 | (n - 2)
    4. D)All of the above
    Show answer

    Answer: All of the above

    Hint: Consider the definition of modular congruence and its relationship to the division algorithm. Are these different ways of saying the same thing?

  4. Q4.medium

    Ravi wants to prove that the number 111111 is a prime number. He checks for divisibility by 2, 3, and 5 and finds that it is not divisible by any of them. He then concludes that 111111 must be a prime number. What is the fundamental flaw in Ravi's approach to determining primality?
    1. A)He should have checked for divisibility by all prime numbers up to the square root of 111111.
    2. B)The number 111111 is actually divisible by 3, making his initial premise false.
    3. C)He only checked the first three prime numbers; a number is prime if it's not divisible by any number other than 1 and itself.
    4. D)He confused the definition of a prime number with that of an odd number.
    Show answer

    Answer: He should have checked for divisibility by all prime numbers up to the square root of 111111.

    Hint: To prove a number is prime, you must systematically check for factors. What is the limit for checking potential prime factors?

  5. Q5.medium

    If 'p' is a prime number, then √p is always a(n) ________ number.
    1. A)rational
    2. B)irrational
    3. C)integer
    4. D)natural
    Show answer

    Answer: irrational

    Hint: Consider the definition of irrational numbers and how square roots of non-perfect squares behave. A prime number is never a perfect square.

  6. Q6.medium

    Which of the following statements about prime numbers is always true?
    1. A)All prime numbers are odd.
    2. B)The product of two distinct prime numbers is always prime.
    3. C)Every composite number can be expressed as a product of prime numbers.
    4. D)The sum of two prime numbers is always an even number.
    Show answer

    Answer: Every composite number can be expressed as a product of prime numbers.

    Hint: Recall the fundamental theorems and definitions related to prime and composite numbers, such as the Fundamental Theorem of Arithmetic.

  7. Q7.hard

    The sum of two positive integers is 100 and their Highest Common Factor (HCF) is 5. How many such distinct pairs of numbers (a, b) exist, where a ≤ b?
    1. A)2
    2. B)3
    3. C)4
    4. D)5
    Show answer

    Answer: 4

    Hint: Express the two numbers in terms of their HCF and two coprime integers. Then use the given sum to find possibilities.

  8. Q8.hard

    Show that for any positive integer n, the expression n³ - n is always divisible by:
    1. A)4
    2. B)5
    3. C)6
    4. D)7
    Show answer

    Answer: 6

    Hint: Factorize the given algebraic expression. What kind of product do you get?

  9. Q9.hard

    What is the remainder when 7¹⁰⁰ is divided by 5?
    1. A)1
    2. B)2
    3. C)3
    4. D)4
    Show answer

    Answer: 1

    Hint: Explore the pattern of the remainders of successive powers of 7 when divided by 5. This is a modular arithmetic problem.

These are 9 of the 40 questions available for Number Theory. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.