Find the sum of those numbers

Find the sum of all positive integers smaller or equal to 2019 which is relatively prime to 2019.


The answer is 1356768.

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2 solutions

Culver Kwan
Jan 22, 2019

If x x is relatively prime to 2019 2019 , 2019 x 2019-x will also be relatively prime to 2019 2019 . There are ϕ ( 2019 ) = 1344 \phi(2019)=1344 numbers relatively prime to 2019 2019 , so there are 1344 2 = 672 \frac{1344}{2}=672 pairs of 201 201 9. So the answer is 2019 × 672 = 1356768 2019 \times 672=1356768 .

( n , 2019 ) = 1 1 n 2019 n = n = 1 2018 n ( 3 × 1 + 3 × 2 + + 3 × 672 ) ( 673 × 1 + 673 × 2 ) = \sum_{(n,2019)=1}^{1 \leq n \leq 2019} n = \sum_{n=1}^{2018}n - (3 \times 1+3 \times 2+\dots + 3 \times 672) - (673 \times 1+ 673 \times 2) =

n = 1 2018 n 3 ( n = 1 672 n ) 673 2 × 673 = 2018 × 2019 2 3 672 × 673 2 673 1346 = 1356768 \sum_{n=1}^{2018}n - 3(\sum_{n=1}^{672}n) - 673-2\times 673=\frac{2018\times 2019}{2}-3\frac{672 \times 673}{2}-673-1346 = 1356768

I have a easier solution which I will post next week.

Culver Kwan - 2 years, 4 months ago

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If it is that easy, why don't you post it now :)

A Former Brilliant Member - 2 years, 4 months ago

Culvar it will be helpful if u post ur solution?

Sefat Bin musa - 2 years, 4 months ago

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I have post my solution already, up there☝️.

Culver Kwan - 2 years, 4 months ago

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