CAT Quantitative Aptitude Questions | CAT Number Systems - Factors

The question is about even factors of a number. A number is given as multiplication of factors. We need to find out the number of even factors. Dealing with factors of a number is a vital component in CAT Number Systems: Factors. A range of CAT questions can be asked based on this simple concept of Factors. CAT exam has always tested the idea of Factors from Number systems. The idea of Factors questions forms an integral part of the CAT syllabus. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score.

Question 6: Number N = 2⁶ * 5⁵ * 7⁶ * 10⁷; how many factors of N are even numbers?

1. 1183
2. 1200
3. 1050
4. 540

Explanatory Answer

Method of solving this CAT Question from Number Theory - Factors: How many even factors does a number have? Now that's a rather odd question. Think about that.

The prime factorization of 2⁶ * 5⁵ * 7⁶ * 10⁷ is 2¹³ * 5¹² * 7⁶.
The total number of factors of N = 14 * 13 * 7
We need to find the total number of even factors. For this, let us find the total number of odd factors and then subtract this from the total number of factors. Any odd factor will have to be a combination of powers of only 5 and 7.
Total number of odd factors of 2¹³ * 5¹² * 7⁶ = (12 + 1) * (6 + 1) = 13 * 7
Total number of factors = (13 + 1) * (12 + 1) * (6 + 1)
Total number of even factors = 14 * 13 * 7 - 13 * 7
Number of even factors = 13 * 13 * 7 = 1183

Alternative approach

Any factor of 2¹³ * 5¹² * 7⁶. will be of the form 2^p * 3^q * 5^r.
Any even factor of 2¹³ * 5¹² * 7⁶ will also be of the same form, except that p cannot be zero in this case.
p can take values 1, 2, 3, .........13 – 13 values
q can take values 0, 1, 2, .........12 – 13 values
r can take values 0, 1, 2, 3,….5 – 7 values
So, the total number of even factors be 13 * 13 * 7 = 1183

The question is "How many factors of N are even numbers?"

Hence the answer is 1183

Choice A is the correct answer.