Question

What is the largest integer n that 4n+1 is a multiple of n+1?

Answer 1

values of n+1
2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,....
multiples of 4n+1
5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,65,69,73,77,81,85,89,93,97,101,...

let 4n+1) = k(n+1)
k = (4n+1)/(n+1) , so k has to be whole number

if n=2 , 9/3 = 3 --> k = 3
if n=3, 13/4 ≠ whole number ,k = (3.25
if n=4, 17/5 ≠ whole number , k = 3.4
if n = 5, 21/6 , no , k = 3.5
if n = 6, 25/7, no , k = 3.57..
if n=7, 29/8 , no , k = 3.625
..
if n = 100, 401/101 = 3.97..
if n = 1 billion, 4,000,000,001/1,000,000,001
= 3.99999..
looks like k ---> 3, but never gets to it
it was 3 when n = 2

can k ever be 4?
check:
4n+1 = 4(n+1)
4n + 1 = 4n + 4
1 = 4, which is not possible, so k can never be 4
could it be 5
4n+1 = 5(n+1)
4n+1 = 5n+5
n = -4 , but n=2 was greater

looks like n=2 was our largest integer value of n