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 2 of 5: Contradiction
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xbc+yca+zab≥3abc−ab−bc−ca>Nxbc+yca+zab\ge3abc-ab-bc-ca>N
Detailed analysis

The bounds from Step 1 imply xbc+yca+zab≥(a−1)bc+(b−1)ca+(c−1)ab=3abc−ab−bc−caxbc+yca+zab\ge(a-1)bc+(b-1)ca+(c-1)ab=3abc-ab-bc-ca. This exceeds NN by abc>0abc>0, contradiction. Thus NN is not representable.