CAT Quantitative Aptitude Questions | CAT Algebra - Polynomials questions

The question is about Factors. We need to find out the factors of the given equation. This questions involves Number Theory. Polynomials is a simple topic which involves a lot of basic ideas. Make sure that you a get of hold of them by solving these questions. A range of CAT questions can be asked based on this simple concept.

Question 8: How many of the following are factors of 3²⁰⁰ – 5¹⁰⁰?
1. 7
2. 16
3. 53
4. 12

1. 3
2. 2
3. 1
4. All of the above

Video Explanation

Explanatory Answer

Method of solving this CAT Question from CAT Algebra - Polynomials: Finding factors of a numbers is uber-easy but that of a term like this is not so easy!

Let us say, N = 3²⁰⁰ – 5¹⁰⁰
N = 9¹⁰⁰ – 5¹⁰⁰.
This is a multiple of 9 - 5. Hence, 4, 2 and 1 are factors of the dividend.
N can also be written as 81⁵⁰ – 25⁵⁰.
This is a multiple of 81 - 25 = 56. Hence, 56, 7, 8 are factors of the dividend.
Dividend = (81²)²⁵ – (25²)²⁵
This is a multiple of 81² - 25², which is equal to (81 - 25) (81 + 25) = 56 × 106 = 7 × 8 × 2 × 53
Hence, 53 is also a factor of the dividend. The ‘2’ from this could combine with the 8 of the previous factor to mean that 16 is also a factor of the dividend.
The one exception of this list is ‘12’. Why? 3²⁰⁰ is a multiple of 3. 5¹⁰⁰ is not a multiple of 3. If a non–multiple of 3 is subtracted from a multiple of 3, the result will be a non–multiple of 3.
Hence, 3 is not a factor of the dividend. => 12 is not a factor of this number.
So, three of the numbers are factors.

The question is "How many of the following are factors of 3²⁰⁰ – 5¹⁰⁰?"

Hence the answer is "3"

Choice A is the correct answer.