# Understand the problem

For each \( n \in \mathbb{N} \) let \( d_n \) denote the G.C.D. of n and (2019 – n). Find the value of \( d_1 + d_2 + … + d_{2019} \).

##### Source of the problem

Regional Math Olympiad, 2019, Maharashtra, Goa Region Problem 1

##### Topic

Number Theory

##### Difficulty Level

5/10

##### Suggested Book

Challenges and Thrills in Pre College Mathematics

# But first, can you try these problems?

- Show that G.C.D. of k and 0 is k for any positive integer k.
- Show rigorously that G.C.D. (a, b) = G.C.D. (a, a+b) for any non-negative integers a and b
- Can you find and prove a similar result with a negative sign?

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#1)Show that G.C.D. of k and 0 is k for any positive integer k.

We know GCD is highest common factor of both k &0;

k=kX1;

0=kX0;

so,GCD=k proved

#2)Show rigorously that G.C.D. (a, b) = G.C.D. (a, a+b) for any non-negative integers a and b;

let GCD of a &b be h;

so a=h*q1; b=h*q2; q1 &q2 do not have any common factor other than 1;or GCD of h1&h2 is 1;

so, (a+b)=h*(q1+q2);

so GCD of a =q1*h& (a+b)=h*(q1+q2);

GCD of a&b=GCD of a&(a+b)=h proved

#3Can you find and prove a similar result with a negative sign?

yes let a=-m=-12(say)& b=-n=-15(say); m &n natural numbers if m&n has GCDas h=3