$Complete \space Question \space is \\ Let \space R \space be \space a \space relation \space on \space the \space set of \space all \space non-negative \space integers \space defined \space by \space aRb \space if \space and \space only \space if \space a ^{ 3 }- b ^{3 } \space is \space divisible \space by \space 6. Then \space relation \space is \space equivalence \space relation .\\ Solution \\ For \space check \space to \space equivalence \space relation, we \space will \space be \space check \\ (1)reflexive, (2) symmetric, and (3) transitive \\ (1) reflexive :-\\ If \space any \space element \space a \space is \space related \space to \space itself.then \space it \space called \space reflexive. Here \space a ^{3 } -a ^{ 3 } \space is \space equal \space to \space zero. And \space zero \space is \space divisible \space by \space 6 \\ So \space say \space that \space aRa. \\ (2)symmetric:- \\ If \space a \space related \space to \space b \space and \space b \space Also \space related \space to \space a, then \space it \space called \space symmetric .\\ Here \\ \space a ^{3 } -b^{ 3 } \space is \space divisible \space by \space 6 \iff \\ \space b ^{3 } -a^{ 3 } \space is \space divisible \space by \space 6 \iff \\ bRa \\ so \space it \space is \space reflexive. \\ (3) transitive:- \\ If \space aRb \space and \space bRc \implies aRc \\ It \space called \space transitive. \\ Here \\ Let \\ \space a ^{3 } -b^{ 3 } \space is \space divisible \space by \space 6 \space and \space\space b^{3 } -c^{ 3 } \space is \space divisible \space by \space 6 \\ \iff [( \space a ^{3 } -b^{ 3 } )+(b^3-c^3)]\space is \space divisible \space by \space 6 \\ \iff \space a ^{3 } -c^{ 3 } \space is \space divisible \space by \space 6 \\ aRc \\ So \space it \space is \space transitive. \\ Finally \space we \space say \space that \space relation \space is \space equivalence \space relation.$