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Let A = n ∈ N : n is a 3 - digit number B = 9 k + 2 : k ∈ N and C = 9 k + l : k ∈ N for some l 0 < l < 9 . If the sum of all the elements of the set A ∩ B ∪ C is 274 × 400 , then l is equal to

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0

Step-by-step solution

B and C will contain three digit numbers of the form 9 k + 2 and 9 k + l respectively. We need to find sum of all elements in the set B ∪ C effectively. Now, S B ∪ C = S B + S C - S B ∩ C where S ( k ) denotes sum of elements of set k . Also, B = 101 , 110 , … … , 992 ∴   S B = 100 2 101 + 992 = 54650 Case-I: If l = 2 then B ∩ C = B ∴   S B ∪ C = S B which is not possible as given sum is 274 × 400 = 109600 . Case-II If l ≠ 2 then B ∩ C

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