Chapter 4.4, Problem 37E

a) Show that ${\text{2}}^{\text{34}0}\equiv \text{1 }\left(\text{mod 11}\right)$ by Fermat's little theorem and noting that ${\text{2}}^{\text{34}0}=\left({\text{2}}^{\text{1}0}\right)\text{34}$ .

b) Show that ${\text{2}}^{\text{34}0}\equiv \text{1 }\left(\text{mod 31}\right)$ using the fact that ${\text{2}}^{\text{34}0}={\left({\text{2}}^{\text{5}}\right)}^{\text{68}}{\text{=32}}^{\text{68}}$ .

c) Conclude from parts (a) and (b) that ${\text{2}}^{\text{34}0}\equiv \text{1}\left(\text{mod 341}\right)$ .