Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7.

Question

Show that the numbers $3$,$3^2$,$3^3$,$3^4$,$3^5$,$3^6$ for a reduced residue system modulo 7.

-A bit lost with this question, we just started a section on reduced residue sets and only covered simple samples in class such as where your number is a prime therefore a complete set of residues is also reduced.

EX: p = 7 : {1,2,3,4,5,6}

Whereas, p = 6 : {1,5} or {7,-1}

Unfortunately these types of examples are the extent of my knowledge so far, any help is appreciated with solving my intial problem.

Answer 1

You are simply asked to compute the remainders of the division by $7$ of these numbers.

You should know that remainders sort of multiply.

So you have

• $3$ gives remainder $3$
• $3^{2}$ gives remainder $2$
• $3^{3}$ gives remainder $2 \cdot 3 = 6$
• $3^{4}$ gives remainder $2^{2} = 4$
• $3^{5}$ gives remainder $5$
• $3^{5}$ gives remainder $1$