Problem Solving

Q)The positive integer 200 has how many factors?

12 or 24??

aditya.j wrote:Q)The positive integer 200 has how many factors?

12 or 24??

First, factor 200 into its primes: 200 = 2*100 = 2*10*10 = 2*2*5*2*5 = 2^3*5^2

Any factor of 200 has to be constructed from these prime factors. That is, it has to be of the form 2^x*5^y where x is 0,1,2, or 3 and y is 0,1 or 2. So, we have 4 choices for the value of x and 3 choices for the value of y, or 4*3=12 total factors.

Ans=12

200=2^3*5^2

no of factors = (3+1)*(2+1) =4*3=12

GmatMathpro

I think factors of 200 should be 24. You have rightly calculated 12 factors but positive, same negative values will also be the factors of 200. SO therefore 12 ( +ve factors ) + 12 ( _ve factors ) == 24 factors.

Please correct me if i m wrong

GmatMathPro wrote:

aditya.j wrote:Q)The positive integer 200 has how many factors?

12 or 24??

First, factor 200 into its primes: 200 = 2*100 = 2*10*10 = 2*2*5*2*5 = 2^3*5^2

Any factor of 200 has to be constructed from these prime factors. That is, it has to be of the form 2^x*5^y where x is 0,1,2, or 3 and y is 0,1 or 2. So, we have 4 choices for the value of x and 3 choices for the value of y, or 4*3=12 total factors.

While considering factors of a number we always count only positive factors.This even applies to HCF and LCM of numbers which can be calculated only for positive numbers.
So answer is 12.

Whats the OA.

ankush123251 wrote:While considering factors of a number we always count only positive factors.This even applies to HCF and LCM of numbers which can be calculated only for positive numbers.
So answer is 12.

200 = 2^3 * 5^2
#factors = (3+1)(2+1) = 12

QA is 12 as i mentioned in my earlier post.

shekhar.kataria wrote:GmatMathpro

I think factors of 200 should be 24. You have rightly calculated 12 factors but positive, same negative values will also be the factors of 200. SO therefore 12 ( +ve factors ) + 12 ( _ve factors ) == 24 factors.

Please correct me if i m wrong

GmatMathPro wrote:

aditya.j wrote:Q)The positive integer 200 has how many factors?

12 or 24??

First, factor 200 into its primes: 200 = 2*100 = 2*10*10 = 2*2*5*2*5 = 2^3*5^2

Any factor of 200 has to be constructed from these prime factors. That is, it has to be of the form 2^x*5^y where x is 0,1,2, or 3 and y is 0,1 or 2. So, we have 4 choices for the value of x and 3 choices for the value of y, or 4*3=12 total factors.

Yea. its funny cos the prep software where i found this sum also said the same thing.

The issue here is that there is no universally accepted definition of 'factor'. For example, Wikipedia says that factors can be positive or negative. On the other hand, Manhattan GMAT's definition of a factor restricts them to positive integers only (p. 15 of their number properties strategies guide). The link here: https://mathforum.org/library/drmath/view/63516.html features mathematicians debating this very question and concluding that there is no consensus.

So, Manhattan and Wikipedia both have legitimate definitions of 'factor', but Manhattan is a GMAT authority, so if they say that factors have to be positive, I would side with them for the purposes of this test. But honestly, if it were a real GMAT question, they would make it very clear in the wording of the question whether the answer should include negatives. There's no way they would write a question where the trick relies on knowing some particular definition of the word 'factor' that's not even universally agreed upon by mathematicians.