Prove that if $n>1$, the sum of positive integers less than $n$ and coprime to $n$ is $(1/2)na(n)$ where $a(n)$ is the number of such integers.

Question

Question 12(iii) Could anyone explain this part of the question to me.

What i tried co-prime means that the two integers a and b are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1

Take th number $3$ for example, then the sum of integers less than $3$ and co-prime to $3$ is $2+1=3$, $2$ and $1$ are the two integers co-prime to $3$ which thus satisfies the formula $0.5*n*a(n)$ where $n=3$, $a(n)=2$. However im unsure of how to prove it. Could anyone explain. Thanks

Hint pair $a$ with its negattion $n-a\equiv -a$. Each pair sums to $n$ and there are ... pairs so.... And don't forget the possibility that $,a\equiv n-a.,$ This is essentially the same as Gauss's trick for summing the fist n integers — Bill Dubuque, Oct 12 '18 at 19:24
Oops, this is the correct dupe link but the idea is the same in both cases. — Bill Dubuque, Oct 12 '18 at 19:28

score 0 · Answer 1 · edited Oct 12 '18 at 21:03

0

Hint:

If $m$ is coprime to $n$ then $n-m$ is also coprime to $n$.

edited Oct 12 '18 at 21:03

dantopa

10,342

answered Oct 12 '18 at 18:57

marty cohen

107,799

Isn't thats what was given in the first part of the question. – Nicole Alison Oct 12 '18 at 18:59

score 0 · Answer 2 · answered Oct 12 '18 at 19:01

0

Hint

You have already shown that $$ (m,n)=1 \iff (m,m-n)=1$$

See if you can figure out what is going on with the sum of such integers.

By the way, the notation for the Euler function is $\phi (n)$ not $\alpha (n)$

answered Oct 12 '18 at 19:01

Mohammad Riazi-Kermani

68,728

Prove that if $n>1$, the sum of positive integers less than $n$ and coprime to $n$ is $(1/2)na(n)$ where $a(n)$ is the number of such integers.

2 Answers2