CAT Practice : Number System - Remainders

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How did Binomial theorem get into Number Theory? More importantly, why did it? Why did the chicken cross the road?

Number Theory - Binomial Theorem

    Q.2: What is the remainder when (13100 + 17100) is divided by 25?
    1. 0
    2. 2
    3. 4
    4. 11

 

  • Correct Answer
    Choice B. 2

Detailed Solution

(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
Answer Choice (B)

Correct Answer: 2

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When we divide 24 pigeons into 5 groups, 4 are left out or remain. This is the remainder. When there is a question based on this idea in the exam, we have a CAT among the pigeons.