Let A = n ∈ N : H . C . F . n , 45 = 1 and let B = 2 k : k ∈ 1 , 2 , … , 100 . Then the sum of all the elements of A ∩ B is _____.

Step-by-step solution

Here B = 2 , 4 , 6 , 8 . . . . , 200 Sum of elements in A ∩ B = 2 + 4 + 6 + … + 200 ⏟ Multiple   of  2 - 6 + 12 + … + 198 ⏟ Multiple   of   2   &  3   i . e .  6 - 10 + 20 + … + 200 ⏟ Multiple   of   5   &  2   i . e .  10 + 30 + 60 + … + 180 ⏟ Multiple   of   2 , 5   &  3   i . e .  30 = 2 100 × 101 2 - 6 33 × 34 2 - 10 20 × 21 2 + 30 6