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?
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
50 can be written
a(b + 1) + b = 50
16 ( 2 + 1) + 2 = 50
48 + 2 = 50
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
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
I found that 46 is the largest number using python, b ut I don't know how to do it mathematically.
But guest, I already showed you that 49 is a solution. So 46 is not the biggest one.
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.)
We're assuming that a, b are not the same integers....( I guess!! )