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

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    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

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