how can we simplify 7^2014 and calculate the remaining when divided by 5?

Dividing by 5 is the same as multiplying by 10 and dividing by 2. Multiplying by 10 just moves the decimal point over one place. Dividing by 2 

7^1 = 7 ÷ 5 = 1.4                7^5 = 16807 ÷ 3361.4

7^2 = 49 ÷ 5 = 9.8              7^6 = 117649 = 23529.8

7^3 = 343 ÷ 5 = 68.6           etc.

7^4 = 2401 ÷ 5 = 480.2

This pattern cycles every 4 numbers. Which row will 7^2014 be in?

Working with the exponents: 1 ÷ 4 = .25           5 ÷ 4 = 1.25

                                        2 ÷ 4 = .50           6 ÷ 4 = 1.50

                                        3 ÷ 4 = .75           7 ÷ 4 =  1.75

                                        4 ÷ 4 =  1.00         8 ÷ 4 = 2.00

2014 ÷ 4 = 503.5   --->   second row 

Therefore, from the top grouping, the decimal is .8 --->  remainder is 5 x .8  = 4

Powers of 7 end with the following digit pattern

7, 9, 3, 1

So 2014 / 4 = 503.5 .......this indicates that 7^2014 ends in a 9

So...the remainder after dividing this by 5 = 4