Real Numbers – important questions for Boards 2027
CBSE Class 10 Maths
Which statement about the prime factorisation of a natural number is correct? (A) It is unique only when the prime factors are written in descending order. (B) It is unique except for the order of its factors. (C) It is never unique if a prime factor occurs more than once. (D) It is unique only for prime numbers.
Answer
(B) It is unique except for the order of its factors — the prime factorisation of a natural number is unique, except for the order of its factors.
State the Fundamental Theorem of Arithmetic, including what it says about the uniqueness of prime factorisation.
Answer
Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
(5 marks) Explain why the prime factorisation of a composite number is unique apart from the order of its prime factors, and state how writing the factors in ascending order expresses this uniqueness. State whether there are finitely or infinitely many primes. Then explain why 4ⁿ cannot end in zero for any natural number n, using prime factorisation.
Answer
The Fundamental Theorem of Arithmetic gives a unique prime factorisation for every composite number, except for the order of the prime factors. 1. Every composite number can be expressed as a product of primes. The factorisation is unique apart from the order in which the prime factors occur. 2. Writing the prime factors in ascending order, p₁ ≤ p₂ ≤ … ≤ pₙ, fixes their order. Once this order is chosen, the factorisation is unique. 3. There are infinitely many primes. Combining repeated prime factors gives powers of primes, such as 32760 = 2³ × 3² × 5 × 7 × 13. 4. For any natural number n, 4ⁿ = 2²ⁿ, so its only prime factor is 2. A number ending in zero is divisible by 5, and its prime factorisation must contain 5; therefore, 4ⁿ cannot end in zero. Thus, unique prime factorisation establishes the result for 4ⁿ.
State how the prime powers are selected when finding the HCF and the LCM of two positive integers.
Answer
For the HCF, select the smallest powers of common prime factors; for the LCM, select the greatest powers of all prime factors involved.
Write the relation connecting the HCF, LCM and the product of two positive integers. Explain what it allows you to find if the HCF and the two integers are known.
Answer
For two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b. If the HCF and both integers are known, divide their product by the HCF to find the LCM.
Let q be a non-zero rational number and x an irrational number. Prove that qx and x/q are irrational. Use a contradiction argument, explaining why assuming either result is rational would conflict with the irrationality of x.
Answer locked. Unlock all answers on SmartKalike
If a prime number p divides a², which conclusion follows? (A) p divides a. (B) a divides p. (C) p² divides a. (D) a² is prime.
Answer locked. Unlock all answers on SmartKalike
When is a real number called irrational? State the condition on the integers in the form p/q used to define a rational number.
Answer locked. Unlock all answers on SmartKalike
Explain how the Fundamental Theorem of Arithmetic is used to study the decimal expansion of a rational number and to prove that numbers such as √2 are irrational. Also state Euclid’s division algorithm.
Answer locked. Unlock all answers on SmartKalike
In the contradiction proof that √3 is irrational, assuming √3 = a/b in lowest terms leads to which conclusion? (A) Neither a nor b is divisible by 3. (B) Both a and b are divisible by 3. (C) a is divisible by 3 but b is not. (D) b is divisible by 3 but a is not.
Answer locked. Unlock all answers on SmartKalike
3 more probable questions in this chapter, every one with a full answer.
Unlock all answers on SmartKalike