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

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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
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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?

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
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50 can be written

a(b + 1) + b  = 50

16 ( 2 + 1) + 2 = 50

48 + 2 =  50   Feb 19, 2019
edited by CPhill  Feb 19, 2019
edited by CPhill  Feb 19, 2019
#3
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Thanks! Do you see any alternative to just brute forcing the solution?

Guest Feb 19, 2019
#5
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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   Feb 19, 2019
edited by CPhill  Feb 19, 2019
edited by CPhill  Feb 19, 2019
edited by 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
#7
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But guest, I already showed you that 49 is a solution.  So 46 is not the biggest one.

Melody  Feb 19, 2019
#8
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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
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See my answer...I believe that it is correct...   CPhill  Feb 19, 2019
#10
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48 can be written with a=6 and b=6, as 6(6+1)+6=48, though

Guest Feb 19, 2019
#11
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We're assuming that a, b  are not the same integers....( I guess!! )   CPhill  Feb 19, 2019
#12
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Thanks guest, you are right, 16  and 2 does work to make 50 and I did read the question incorrectly

Me You Melody  Feb 19, 2019
edited by Melody  Feb 19, 2019
#13
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Oh,  I'm pretty sure they can be though :/

Guest Feb 19, 2019
#14
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So it seems like the only way to solve this is with brute force?

Guest Feb 19, 2019