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?

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06-03-2017
Mathematics

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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?

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LammettHash LammettHash · Answer 1 · 2017-03-06T21:30:16+01:00

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

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

The sum of these multiples is

[tex]\displaystyle\sum_{k=1}^{191}7k+\sum_{k=1}^{121}11k-\sum_{k=1}^{17}77k[/tex]

which can be computed using the well-known formula,

[tex]\displaystyle\sum_{k=1}^nk=\dfrac{n(n+1)}2[/tex]

So you have

[tex]\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[/tex]