Find the identity element of S with respect to $*$.

Show that every element of S has an inverse with respect to $*$.

State which elements of S are self-inverse with respect to $*$.

Prove that the operation $\circ$ is not distributive over $*$.

Determine the equivalence classes into which R partitions S, giving the first four elements of each class.

Find two elements in the same equivalence class which are inverses of each other with respect to $*$.

$a + e + 20 = a$(mod 100)     (M1)

$e =  - 20$(mod 100)       (A1)

$e = 80$      A1

[3 marks]

$a + {a^{ - 1}} + 20 = 80$(mod 100)     (M1)

inverse of $a$ is $60 - a$ (mod 100)        A1

[2 marks]

30 and 80       A1A1

[2 marks]

$a \circ \left( {b * c} \right) = a \circ \left( {b + c + 20} \right)$(mod 100)

$= a + \left( {b + c + 20} \right) - 20$(mod 100)      (M1)

$= a + b + c$(mod 100)      A1

$\left( {a \circ b} \right) * \left( {a \circ c} \right) = \left( {a + b - 20} \right) * \left( {a + c - 20} \right)$(mod 100)      M1

$= a + b - 20 + a + c - 20 + 20$(mod 100)

$= 2a + b + c - 20$(mod 100)      A1

hence we have shown that $a \circ \left( {b * c} \right) \ne \left( {a \circ b} \right) * \left( {a \circ c} \right)$      R1

hence the operation $\circ$ is not distributive over $*$      AG

Note: Accept a counterexample.

[5 marks]

{0,5,10,15...}      A1

{1,4,11,14...}      A1

{2,3,12,13...}      A1

{6,9,16,19...}      A1

{7,8,17,18...}      A1

[5 marks]

for example 10 and 50, 20 and 40, 0 and 60…     A2

[2 marks]