What is the largest integer that cannot be expressed in the form 2 4 a + 7 b where a and b are non-negative integers?
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aka Chicken McNugget Theorem :) See link from AOPS
By the Chicken McNugget Theorem , the answer is 2 4 ⋅ 7 − 2 4 − 7 = 1 3 7 .
We have to find THE FOBRENIUS NUMBER.........Which is equivalent to.... {[a-d][b-d]/d}-d........where gcd(a,b) is d......and ax+ by....therefore.....137!
Well I'll tell you a thing..for every g cd ( x , y ) = 1
a x + b y = ⋯ m a n y n u m b e r s . . . [ x , y ∈ Z + (I am not sure on this) ] also not to forget a , b ∈ n o n . n e g a t i v e Z
the total no. of numbers which cannot be expressed in this form are
2 ( x − 1 ) ( y − 1 ) ...now here I can't have to prove it
and the highest of them..is ( x − 1 ) ( y − 1 ) − 1 ...
now get it..
It is known as the Fobrenius number
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This is a version of "The Coin Problem".
(For a discussion see here .)
Since 2 4 and 7 are relatively prime, the solution is
2 4 ∗ 7 − 2 4 − 7 = 1 3 7 .