Number of Factors > 1 & Repeating Decimals

GRE General
1

Hi all,

I have a couple of questions :

  1. I know how to calculate the number of factors of an integer by first finding the prime factors, adding 1 to the exponents of the prime factors, and then multiplying the exponents. However, there were a couple of questions in the Manhattan 5lb book which asked to find the number of factors greater than 1. How do we use the above method to do this?

  2. When it comes to questions with repeating decimals is there a specific strategy to follow e.g. breaking it apart into a constant and the repeating part? In this question (screenshot attached) I felt that there was a lot of abstract manipulation that was taking place.

    Screenshot 2023-05-23 at 8.06.01 PM
    Screenshot 2023-05-23 at 8.06.01 PM1914×988 58.7 KB

2

Regarding Q2 you can see there is a clear denominator hint 5.0001 how can it be written 5+ (110^-4) and numerator can be written in form of 25 - 110^-8 basically the question has been given in form of (a^2 - b^2)(a-b) form

Now regarding Q1 I could nott get your question if suppose the number of factors is 24 , The number of factors greater than 1 would be 24-1 which is 23 , Is this what you are asking ? @sarahthussain

3

Thank you so much for your response.

Re Q2 I guess I see it now after your’s and greg’s explanations but I wouldn’t have been able to come up with it on my own… I guess with more practice I will be able to see patterns.

Re Q1, ah yes that makes a lot of sense! In the method I described we add 1 to the exponents to account for the possibility of the exponent being 0 and therefore 1 being a possible factor. However, we add 1 to all the exponents so aren’t we double-counting the 1s in a way? Perhaps, I don’t fully understand why we are adding 1 to the exponents. Would you be able to explain that as well :stuck_out_tongue:

4

@sarahthussain Actually I have no idea about how the formula of (a+1)(b+1)(c+1) is derived , But one thing is for sure in that , the number 1 is a part of the factors

Take example number of 30
Using the formula - Number can be written as 235
Number of factors by exponents of prime numbers (2)(2)(2) = 8
Factors = 1,2,3,5,6,10,15,30 (8 Numbers)

1 is also part of this formula
So whenever they ask for number of factors excluding 1 I think best option is to do prime factorization and subtract 1 from the formula to get the answer