Please show all steps involved and as clearly to read as pos

Please show all steps involved and as clearly to read as possible.

Solution

Since it is given in the problem that a^6 = e, then the maximum value of |a| will be 6, hence we left with the values as 1,2,3,4,5,6

If a^4 = e, then we can a^2 = a^2e = a^2*a^6 = (a^4)^2 = e^2 = e, so the value of |a| =4 is an impossible value

We get the same result for |a| equal to 5

a^3 = a^3e^2 = a^3(a^6)^2 = a^(15) = (a^5)^3 = e^3 = 3, hence |a| = 5 is also an impossible values

So what are the possible values left for |a| are

{1,2,3,4,5,6} - {4,5} = {1,2,3,6}

Hence there are only 4 possible values which are 1,2,3 and 6