Real Numbers

Define a prime number and give an example of the smallest prime number.

State the Fundamental Theorem of Arithmetic.

Find the HCF of 6 and 20 by prime factorization method.

Verify that for any two positive integers a and b, HCF(a, b) × LCM(a, b) = a × b, taking a = 6 and b = 20.

Find the HCF and LCM of 12, 15 and 21 using the prime factorization method.

Given that HCF(306, 657) = 9, find LCM(306, 657).

Check whether 6ⁿ can end with the digit 0 for any natural number n.

Explain why the number √2 cannot be expressed as a rational number.

Prove that √5 is an irrational number.

Prove that 3 + 2√5 is an irrational number, given that √5 is irrational.

Without actually performing the long division, state whether the rational number 13/3125 has a terminating or a non-terminating repeating decimal expansion.

State the Fundamental Theorem of Arithmetic.

Explain why $7 \times 11 \times 13 + 13$ is a composite number.

Explain Euclid's Division Lemma in detail.

Explain why the number $4^n$ cannot end with the digit zero for any natural number $n$.

Find the HCF and LCM of 6 and 20 by the prime factorization method.

Prove that $\sqrt{2}$ is an irrational number by explaining the core concept.

Find the HCF of 96 and 404 by using the prime factorization method.

Find the LCM of 72 and 120 using prime factorization.

Find the LCM and HCF of 336 and 54 by the prime factorization method, and verify that $\text{LCM} \times \text{HCF} = \text{Product of the two numbers}$.

Find the HCF and LCM of 336 and 54 by the prime factorization method, and verify that $\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$.

Find the HCF and LCM of 96 and 404 by the prime factorization method and verify that HCF × LCM = product of the two numbers.

Explain the Fundamental Theorem of Arithmetic in detail. Discuss its uniqueness and its various applications in the field of mathematics, specifically regarding the properties of numbers.

Prove that $\sqrt{3}$ is an irrational number using the method of contradiction.

Explain the Fundamental Theorem of Arithmetic in detail. Discuss its significance in number theory and describe at least three of its major applications in mathematics. (5 Marks)

(a) Prove that $\sqrt{3}$ is an irrational number. (b) Find the HCF and LCM of 96 and 404 by the prime factorization method and verify that $HCF \times LCM = \text{Product of the two numbers}$. (5 Marks)