Given four integers, show that the product of the six differences is divisible by 12.
Solution
Two of the integers must have the same residue mod 3 and hence their difference must be a multiple of 3. Similarly, two of the integers, say A and B, must have the same parity and hence their difference is even. Now either one of the other two numbers, say C, has the same parity as A in which case A-C is even. Or both the other two numbers have opposite parity to A, in which case they have the same parity as each other and hence their difference is even.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03