Answer to Question #350944 in Combinatorics | Number Theory for WSN

Question #350944

1. Find all positive integers n such that n2+ 1 is divisible by n+ 1. 


1
Expert's answer
2022-06-26T04:17:43-0400

gcd(n+1,n²+1) = gcd(n+1,n²+1-n(n+1))= gcd(n+1,1-n) =gcd(n+1,1-n+(1+n))=gcd(n+1,2).


If n is even , then n+1 is odd. Then gcd(n+1,2) =1, hence


gcd(n+1,n²+1)=1 so that n+1, n²+1 are relative primes.


If n is odd, then n+1 is even , so gcd(n+1,2)=2, thus gcd(n+1,n²+1)=2. Remember that: if a>0, then

gcd(a,b)=a"\\iff" a|b.

Then n+1|n²+1 "\\iff" gcd(n+1,n²+1)=n+1,

then n+1 must be equal to 2, then n=1 is the only positive integer that satisfies the property.


Answer: 1

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