MathCastle
Groups and FieldsMediumGroups and the Group Axioms

Groups and the Group Axioms · Medium problem

For k=1,2,3,4k=1,2,3,4, define a∗b=(ka+b) mod 5a*b=(ka+b)\bmod5 on {0,1,2,3,4}\{0,1,2,3,4\}. For how many of these four values of kk is ∗* commutative (does a∗b=b∗aa*b=b*a hold for every a,ba,b)?