Quantity-I: 'Arjun' and 'Bheem' are running on a circular track with a diameter of 'a' meters, at speeds of 25 m/s and 40 m/s, respectively. If they are moving in the same direction, determine the number of distinct points where they will meet. Quantity-II: Two different numbers have an LCM of 924 and an HCF of 11. Determine the total number of such possible pairs.

ATQ, Quantity I: Ratio of speeds of 'Arjun' and 'Bheem' = 25:40 = 5:8 We know that when two people are racing on a circular track at 'x' m/s and 'y' m/s in same direction, the number of distinct meeting points will be |x - y|, where 'x' and 'y' are co-prime numbers. So, number of distinct meeting points = 8 - 5 = 3 So, Quantity I = 3 Quantity II: Let the two numbers be '11x' and '11y' where 'x' and 'y' are co-primes. So, 11x × 11y = 924 × 11 Or, xy = 84 So, xy = 84 = 2² × 3 × 7 If N = x^p × y^q X z^r....., then, the number of ways of writing 'N' as a product of 2 co-primes is 2^{(n - 1)}, where 'n' is the number of distinct prime factors of the given number N. Number of ways in which 84 can be written as a product of two co-primes = 2 ^{(3 - 1)} = 4 So, Quantity II = 4 So, Quantity I < Quantity II