Find the HCF and LCM of 306 and 54 . Verify that HCF × LCM = Product of the two numbers

Given numbers are: 306 , 54 . Applying the prime factorisation method to find the prime factors. 306 = 2 × 3 × 3 × 17 = 2 × 3 2 × 17 , and 54 = 2 × 3 × 3 × 3 = 2 × 3 3 . ∴ H . C . F . of 306 , 54 = 2 × 3 2 = 18 , and L . C . M . of 306 , 54 = 2 × 3 3 × 17 = 918 . The statement to verify is: HCF × LCM = Product of the two numbers L . H . S . = HCF × LCM = 18 × 918 = 16524 R . H . S = 306 × 54 = 16524 L . H . S . = R . H . S . Therefore, the statement is verified.

Prove that 1 2 + 3 is an irrational number.

Let us assume that 1 2 + 3 is a rational number. Now, let 1 2 + 3 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 2 + 3 = b a ⇒ 3 = b a - 2 Since a , b are rational numbers, then b a - 2 is also a rational number. ⇒ 3 is a rational number But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 1 2 + 3 is an irrational number.

Prove that 3 + 5 is an irrational number.

Assume that 3 + 5 is a rational number equal to ' a ' . ⇒ a = 3 + 5 . Squaring both sides, ⇒ a 2 = 3 + 5 2 ⇒ a 2 = 3 + 5 + 2 3 5 ⇒ a 2 = 8 + 2 15 ⇒ 2 15 = a 2 - 8 ⇒ 15 = a 2 - 8 2 Now, a 2 - 8 2 is a rational number, since a is a rational number. ⇒ 15 is a rational number. But, we know that, 15 is an irrational number. Hence, our assumption is wrong. Therefore, 3 + 5 is an irrational number.

Prove that 2 3 - 7 is an irrational.

Let us assume that 2 3 - 7 is a rational number. Now, let 2 3 - 7 = a b , where a , b are co-primes and b ≠ 0 . ⇒ 2 3 = a b + 7 ⇒ 3 = 1 2 a b + 7 Since a , b are integers, then a b + 7 is also a rational number. ⇒ 1 2 a b + 7 is a rational number. ⇒ 3 is a rational number. But, we know that, 3 is an irrational number. This shows our initial assumption was incorrect. Therefore, 2 3 - 7 is an irrational number.