CAT Quantitative Aptitude Questions | CAT Number Systems - Remainders

The question is from CAT Remainders: Binomial theorem. A number is given in (x^a + y^a). 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

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.