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What is the greatest integer x, where \(0\leq x\leq50 \) that cannot be written as a(b+1)+b, where a and b are positive integers?

Guest Feb 19, 2019

#1**+1 **

What is the greatest integer x, where \(0\leq x\leq50\) that cannot be written as a(b+1)+b, where a and b are positive integers?

Well if the numbers are 1 and 24 then x will be **49**.

I do not see how to get 50.

So maybe the biggest x value is 49

Melody Feb 19, 2019

#5**+2 **

49 = 9*4 + 9 + 4

I don't believe that 48 can be written

Let's suppose that it can.....we have

a( b+ 1) + b = 48

ab + a + b = 48

ab + b = 48 - a

b ( a + 1) = 48 - a

b = 48 - a

______

a + 1

"a" cannot be odd because we would have odd / even = no integer possible for "b"

Possible values for a, b

a b

0 48 not possible.....a,b must be positive

6 6 not unique for a, b

And no "a" from 8 - 22 inclusive will produce an integer

And if a > 22 and even....then we will have a proper fraction for "b".... [ not an integer ]

So.....48 is the largest integer that cannot be written

CPhill Feb 19, 2019

#6**0 **

I found that 46 is the largest number using python, b ut I don't know how to do it mathematically.

Guest Feb 19, 2019

edited by
Guest
Feb 19, 2019

edited by Guest Feb 19, 2019

edited by Guest Feb 19, 2019

#7**-1 **

But guest, I already showed you that 49 is a solution. So 46 is not the biggest one.

Melody
Feb 19, 2019

#8**+1 **

Sorry, but my question was numbers that CANNOT be written as such, but thank you anyways. :)

(50 can be written as 2(16+1)+16=50 as my progam found.)

Guest Feb 19, 2019

edited by
Guest
Feb 19, 2019