Number Theory : Test of Divisibility Question

Divisibility of a large number

Question

The largest number amongst the following that will perfectly divide 101¹⁰⁰ - 1 is

1. 100
2. 10,000
3. 100¹⁰⁰
4. 100,000
   

Correct Choice is (2) and Correct Answer is 10,000

Explanatory Answer

The easiest way to solve such problems for CAT puposes is trial and error or by back substituting answers in the choices given.

101² = 10201.
101² - 1 = 10200. This is divisible by 100.

Similarly try for 101³ - 1 = 1030301 - 1 = 1030300.

So you can safely conclude that (101¹ - 1) to (101⁹ - 1) will be divisible by 100.
(101¹⁰ - 1) to (101⁹⁹ - 1) will be divisible by 1000.
Therefore, (101¹⁰⁰ - 1) will be divisible by 10,000.

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