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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?

 Feb 19, 2019
 #1
avatar+100819 
+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

 Feb 19, 2019
 #2
avatar+100570 
+3

50 can be written 

 

a(b + 1) + b  = 50

 

16 ( 2 + 1) + 2 = 50

 

48 + 2 =  50

 

 

cool cool cool

 Feb 19, 2019
edited by CPhill  Feb 19, 2019
edited by CPhill  Feb 19, 2019
 #3
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0

Thanks! Do you see any alternative to just brute forcing the solution?

Guest Feb 19, 2019
 #5
avatar+100570 
+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

 

 

cool cool cool

 Feb 19, 2019
edited by CPhill  Feb 19, 2019
edited by CPhill  Feb 19, 2019
edited by CPhill  Feb 19, 2019
 #6
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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
 #7
avatar+100819 
-1

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

Melody  Feb 19, 2019
 #8
avatar
+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
 #9
avatar+100570 
+1

See my answer...I believe that it is correct...

 

cool cool cool

CPhill  Feb 19, 2019
 #10
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0

48 can be written with a=6 and b=6, as 6(6+1)+6=48, though

Guest Feb 19, 2019
 #11
avatar+100570 
+2

We're assuming that a, b  are not the same integers....( I guess!! )

 

cool cool cool

CPhill  Feb 19, 2019
 #12
avatar+100819 
-1

Thanks guest, you are right, 16  and 2 does work to make 50 and I did read the question incorrectly 

 

Me  angry 

You  cool 

Melody  Feb 19, 2019
edited by Melody  Feb 19, 2019
 #13
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0

Oh,  I'm pretty sure they can be though :/

Guest Feb 19, 2019
 #14
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0

So it seems like the only way to solve this is with brute force?

Guest Feb 19, 2019

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