CAT Quantitative Aptitude Questions | CAT Number Systems - Remainders

CAT Questions | Number Theory | Remainders - Binomial theorem

The question is from CAT Remainders: Binomial theorem. A number is given in (xa + ya). We need to find out the remainder when it is divided by b. A range of CAT questions can be asked based on concepts from Remainders. How did Binomial theorem get into Number Theory? More importantly, why did it? Why did the chicken cross the road? :).

Question 2: What is the remainder when (13100 + 17100) is divided by 25?

  1. 0
  2. 2
  3. 4
  4. 11

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Explanatory Answer

Method of solving this CAT Question from Number Theory - Remainders: How did Binomial theorem get into Number Theory? More importantly, why did it? Why did the chicken cross the road?

(13100 + 17100) = (15 – 2)100 + (15 + 2)100
Now 52 = 25, So, any term that has 52 or 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 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

The question is "What is the remainder when (13100 + 17100) is divided by 25?"

Hence the answer is "2".

Choice B is the correct answer.

 

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