SOLUTION: what is the remainder when 5^2 + 5^3 + 5^4 + .......+ 5^247 is divided by 52

52+53+54+...+5247 =

(52+53+54+55)+54(52+53+54+55)+58(52+53+54+55)+512(52+53+54+55)+ ... +5240(52+53+54+55)+5244(52+53) =

(3900)+54(3900)+58(3900)+512(3900)+ ... +5240(3900)+5244(150)

Since 3900 is divisible by 52 we only need to look at the
remainder when 5244(150) is divided by 52


5244(150) = (5244-1+1)(150) = (5244)(150)-1(150)+1(150) = 150[(5244)-1]+150=150[(54)61-1] + 150

Since xk-1 is divisible by x-1,

(54)61-1 is divisible by 54-1 or 624 which is divisible by 52, so

we only need to find the remainder when 150 is divided by 52

     2
52)150
   104
    46
Answer: 46

Edwin