Question

4. Find the sum of all the numbers from 100 to 199 that are not divisible by 13.

What is the formula for
this?

Answer 1

There are 100 numbers in the range 100-199. Find the smallest and largest multiple of 13 in this range. We have

104 = 8•13

195 = 15•13

so there are 15 - 8 + 1 = 8 multiples of 13 between 100 and 199.

Then the sum we want is

`\displaystyle \sum_(k=100)^(199) k - \sum_(\ell=8)^(15)13\ell`

or equivalently,

(100 + 101 + 102 + … + 199) - 13 (8 + 9 + … + 15)

To compute these sums, recall the following formula:

`\displaystyle \sum_(i=1)^n i = 1 + 2 + 3 + \cdots + n = \frac{n(n+1)}2`

Then

`\displaystyle \sum_(k=100)^(199) k = \sum_(k=1)^(199) k - \sum_(k=1)^(99) k = \frac{199\cdot200}2 - \frac{99\cdot100}2 = 14,950`

and

`\displaystyle \sum_(\ell=8)^(15) 13\ell = 13 \left(\sum_(\ell=1)^(15) \ell - \sum_(\ell=1)^7 \ell\right) = 13 \left(\frac{15\cdot16}2 - \frac{7\cdot8}2\right) = 1,196`

which means the sum we want is 14,950 - 1,196 = 13,754.