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 4 of 5: Apply the two-number theorem
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m>bc−b−cm>bc-b-c
Detailed analysis

Since x≤a−1x\le a-1 and n>Nn>N, n−xbc>abc−ab−acn-xbc>abc-ab-ac, hence m>bc−b−cm>bc-b-c. As gcd⁡(b,c)=1\gcd(b,c)=1, every integer greater than bc−b−cbc-b-c has the form m=yc+zbm=yc+zb with y,z≥0y,z\ge0.