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Old 2010-04-15, 01:20   #1
wreck
 
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"Bo Chen"
Oct 2005
Wuhan,China

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Default Ask a number theory question

This week I meet a problem that can't be solved by myself totally, so here it is.

Let a, b be two positive integers, a<b.
Then what's the smallest number could be expressed as i*a+j*b, where i and j are integers, i>-1,j>-1.

For example, if a=3,b=4,then c=6,since 5 can not be expressed as i*3+j*4 and 6=2*3+0*4,7=1*3+1*4,8=0*3+2*4,9=3*3+0*4,etc.

And Then if a is an even number, b=a+1, it seems like c=a^2-a, but I don't know how to proof it. Another question is if a is an odd number and b=a+2,how to compute c and is there a simple formula to express c?

Thanks for your attention.

--Bo Chen--
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Old 2010-04-15, 05:01   #2
jyb
 
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Aug 2005
Seattle, WA

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Quote:
Originally Posted by wreck View Post
This week I meet a problem that can't be solved by myself totally, so here it is.

Let a, b be two positive integers, a<b.
Then what's the smallest number could be expressed as i*a+j*b, where i and j are integers, i>-1,j>-1.

For example, if a=3,b=4,then c=6,since 5 can not be expressed as i*3+j*4 and 6=2*3+0*4,7=1*3+1*4,8=0*3+2*4,9=3*3+0*4,etc.

And Then if a is an even number, b=a+1, it seems like c=a^2-a, but I don't know how to proof it. Another question is if a is an odd number and b=a+2,how to compute c and is there a simple formula to express c?

Thanks for your attention.

--Bo Chen--
I think you need to revise your statement of the problem. The way you've posed it, I believe the answer is always i = j = 0 => c = 0.
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Old 2010-04-15, 05:41   #3
Zeta-Flux
 
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May 2003

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I think the problem is supposed to be: find the *largest* number which is *not* expressible in that form. This is often called the Chicken McNugget Theorem
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Old 2010-04-15, 07:05   #4
wreck
 
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"Bo Chen"
Oct 2005
Wuhan,China

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Quote:
Originally Posted by jyb View Post
I think you need to revise your statement of the problem. The way you've posed it, I believe the answer is always i = j = 0 => c = 0.
I'm sorry, it is indeed that I only want to find the positive value of c and (a,b)=1 is the condition.
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Old 2010-04-15, 07:11   #5
wreck
 
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"Bo Chen"
Oct 2005
Wuhan,China

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Quote:
Originally Posted by Zeta-Flux View Post
I think the problem is supposed to be: find the *largest* number which is *not* expressible in that form. This is often called the Chicken McNugget Theorem
Oh, that is indeed what I want, so from the url what I want is c=ab-a-b+1.

Thanks very much.
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