Question

If we list all the natural numbers below 20 that are multiples of 7 or 11, we get 7, 11 and 14. The sum of these multiples is 32. What is the sum of all the multiples of 7 or 11 up to and including 1337?

Answer 1

Number of multiples of 7 up to 1337: $\left\lfloor\dfrac{1337}7\right\rfloor=191$
Number of multiples of 11 up to 1337: $\left\lfloor\dfrac{1337}{11}\right\rfloor=121$
Number of multiples of 77 up to 1337: $\left\lfloor\dfrac{1337}{77}\right\rfloor=17$

This means there are $191+121-17=295$ distinct multiples of 7 *or* 11 up to 1337.

The sum of these multiples is

$\displaystyle\sum_{k=1}^{191}7k+\sum_{k=1}^{121}11k-\sum_{k=1}^{17}77k$

which can be computed using the well-known formula,

$\displaystyle\sum_{k=1}^nk=\dfrac{n(n+1)}2$

So you have

$\displaystyle\sum_{k=1}^{191}7k+\sum_{k=1}^{121}11k-\sum_{k=1}^{17}77k=7\dfrac{191\times192}2+11\dfrac{121\times122}2-77\dfrac{17\times18}2=197762$