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 3 of 5: Choose a residue for a larger nn
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0≤x<a,xbc≡n(moda)0\le x<a,\qquad xbc\equiv n\pmod a
Detailed analysis

Let n>Nn>N. Because bcbc is invertible modulo aa, choose the unique xx with 0≤x<a0\le x<a and xbc≡n(moda)xbc\equiv n\pmod a. Then m=(n−xbc)/am=(n-xbc)/a is an integer.