Consider the sequence of natural numbers from 1 to 140 inclusive. The sum of all numbers in this sequence that share no common prime factor with 70 is

Step-by-step solution

The prime factorization of 70 is 2 5 7 . We need to find the sum of all natural numbers up to 140 that are not divisible by 2 , 5 , or 7 . Total sum of numbers from 1 to 140 is S = 140 141 2 = 9870 . Let S_k denote the sum of multiples of k up to 140 . S₂ = 70 2 (2 + 140) = 35 142 = 4970 S₅ = 28 2 (5 + 140) = 14 145 = 2030 S₇ = 20 2 (7 + 140) = 10 147 = 1470 S₁₀ = 14 2 (10 + 140) = 7 150 = 1050 S₁₄ = 10 2 (14 + 140) = 5 154 = 770 S₃₅ = 4 2 (35 + 140) = 2 175 = 350 S₇₀ = 2 2 (70 + 140) = 210 By the Principle of Incl