Number Theory

Question

What is the remainder when (13¹⁰⁰ +1 7¹⁰⁰) 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 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