A set of m parallel lines intersects another set of n parallel lines, where m > n . If the total number of parallelograms formed by these lines is 420 , then the maximum number of intersection points between these two sets of lines is

Step-by-step solution

The number of parallelograms formed by the intersection of a set of m parallel lines and a set of n parallel lines is given by choosing 2 lines from the first set and 2 lines from the second set. Number of parallelograms = ^ m C₂ ^ n C₂ = 420 m(m-1) 2 n(n-1) 2 = 420 m(m-1)n(n-1) = 1680 Prime factorization of 1680 gives 1680 = 2^4 3 5 7 . We can group these factors into products of consecutive integers: 1680 = 56 30 = (8 7) (6 5) Since m > n , we can match the terms to get m = 8 and n = 6 . The maximum number of int