Problem 3

Let

a,b,c

be positive integers, no two having a common divisor greater than

1

. Show that

2abc-ab-bc-ca

is the largest integer which cannot be expressed as

xbc+yca+zab

, where

x,y,z

are non-negative integers.

Step 1 of 5: Assume

N

is representable

Previous step Next step

N=2abc-ab-bc-ca\equiv -bc\pmod a

Detailed analysis

Suppose $N=xbc+yca+zab$ . Reducing modulo $a$ gives $xbc\equiv-bc\pmod a$ . Since $\gcd(a,bc)=1$ , we obtain $x\equiv-1\pmod a$ , so $x\ge a-1$ . Cyclically, $y\ge b-1$ and $z\ge c-1$ .