For integers a b c if abc 1 mod 3 prove that either ONE or

For integers a, b, c, if abc ? 1 (mod 3), prove that either ONE or ALL THREE of a, b, c must be
congruent to 1 modulo 3

Solution

Let abc is congruent to 1 mod 3.

If possible let a , b , c have difference module

a =0, b =1 and c =2

Then abc = multiple of 3, hence abc cannot have 0 remainder

Case II: a=1, b =1 and c =2

Then abc = (3l+1)(3m+1)(3n+2) = 27lmn+9l+3l+3m+...+2

Of these all will be multiples of 3 except 2

Hence abc will have 2 mod 3

contradiction

Hence if abc = 1 mod3 then a, b, c individually should have 1 mod 3.