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What is the remainder when (13100 + 17100) is divided by 25? Notes - Class 12

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(13100 + 17100) = (15 – 2)100 + (15 + 2)100
Now 5= 25, So, any term that has 5or 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 150 or 151.
(15 – 2)100 + (15 + 2)100
Coefficient of 15= (-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 2by 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
Hence the answer is "2". 

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