MathLabs

Problem 3

Let a,b,ca,b,c be positive integers, no two having a common divisor greater than 11. Show that 2abc−ab−bc−ca2abc-ab-bc-ca is the largest integer which cannot be expressed as xbc+yca+zabxbc+yca+zab, where x,y,zx,y,z are non-negative integers.
Step 5 of 5: Represent every larger integer
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n=xbc+yca+zabn=xbc+yca+zab
Detailed analysis

Substituting m=yc+zbm=yc+zb into n=xbc+amn=xbc+am gives n=xbc+yca+zabn=xbc+yca+zab. Thus every n>Nn>N is representable, while Step 2 proves NN is not; therefore NN is the largest nonrepresentable integer.