Let the two numbers be \( x \) and \( x + 10 \), as one number exceeds the other by 10.
The sum of the numbers is 40: \[ x + (x + 10) = 40 \quad \Rightarrow \quad 2x + 10 = 40 \quad \Rightarrow \quad 2x = 30 \quad \Rightarrow \quad x = 15 \] The two numbers are 15 and 25. To find the LCM of 15 and 25, we use the prime factorization method: \[ 15 = 3 \times 5, \quad 25 = 5 \times 5 \] The LCM is the product of the highest powers of all prime factors: \[ LCM = 3 \times 5^2 = 75 \] To find the LCM, use the prime factorization of the numbers and take the highest powers of all primes.
Directions: In Question Numbers 19 and 20, a statement of Assertion (A) is followed by a statement of Reason (R).
Choose the correct option from the following:
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.
Assertion (A): For any two prime numbers $p$ and $q$, their HCF is 1 and LCM is $p + q$.
Reason (R): For any two natural numbers, HCF × LCM = product of numbers.