Let S₁ = x Z : 1 x 600, x = 5k + 2 for some k Z , S₂ = x Z : 1 x 600, x = 7k + 3 for some k Z , and S₃ = x Z : 1 x 600, x = 6k + 5 for some k Z . The sum of the elements in the set S₁ S₂ S₃ is equal to _____.

Step-by-step solution

The elements of the sets can be viewed as terms of arithmetic progressions. For S₁ , the terms are of the form x = 5k + 2 , which means x 2 5 . For S₂ , the terms are of the form x = 7k + 3 , which means x 3 7 . For S₃ , the terms are of the form x = 6k + 5 , which means x 5 6 . We first find the smallest positive integer that satisfies all three conditions. From x = 5k + 2 , we substitute into the second condition: 5k + 2 3 7 5k 1 7 k 3 7 . Thus, k = 7m + 3 , and substituting this back gives x = 5(7m + 3) + 2 = 35