 # Number Theory

Question
What is the remainder when (13100 +17100) is divided by 25?
A. 2
B. 0
C. 15
D. 8
Correct Answer: Choice (A)
Explanation:
What is the remainder when (13100 +1 7100) is divided by 25?
(13100 +1 7100) = (15 – 2)100+ (15 + 2)100
Now 52 = 25, So, any term that has 52or any higher power of 5 will be a multiple of 25. So, for the above question, for computing remainder, we need to think about only the terms with 150or 151.
(15 – 2)100 + (15 + 2)100
Coefficient of 150 = (-2)100 + 2100
Coefficient of 151 = 100C1* 151* (-2)99 + 100C1 * 151* (-2)99 . These two terms cancel each other.So, the sum is 0.
Remainder is nothing but (-2)100 + 2100 =(2)100 + 2100
2101
Remainder of dividing 21 by 25 = 2
Remainder of dividing 22 by 25 = 4
Remainder of dividing 23 by 25 = 8
Remainder of dividing 24 by 25 = 16
Remainder of dividing 25 by 25 = 32 = 7
Remainder of dividing 210 by 25 = 72  = 49 = -1
Remainder of dividing 220 by 25 = (-1)2  = 1
Remainder of dividing 2101 by 25 = Remainder of dividing 2100 by 25 * Remainder of dividing 21 by 25 = 1 * 2 = 2