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Sum(not div by 7) = Sum(all) - Sum(div by 7)

Now the sum of an AP is

Sn = n/2 (first + last)

where n is the number of terms

Sum(All) = 10000/2 (1 + 10000) = 50005000

Sum(div by 7) = (9996/7)/2 (7 + 9996) = 1428/2 (10003) = 7142142

Sum(not div by 7) = Sum(all) - Sum(div by 7)

= 50005000 - 7142142

= 42 862 858

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Q: Find the sum of all positive integers not greater than 10000 that are not divisible by 7?
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This is easiest to answer by summing all the numbers 1-10000 and subtracting the sum of the multiples of 7 (7, 14, 21, ..., 9996). The sum of a series is: S = (first + last) x number_of_terms / 2 For for 1-10000, the sum is: S1 = (1 + 10000) x 10000 / 2 = 10001 x 5000 = 50005000 For the multiples of 7 the sum is: S2 = (7 + 9996) x 1428 / 2 = 10003 x 714 = 7142142 So the sum of all integers not greater than 10000 that are not divisible by 7 is: S = S1 - S2 = 50005000 - 7142142 = 42,862,858


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