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Magoosh Brain Twister: Factorial Fun – Explanation


Welcome back! If you haven’t tried your luck with this week’s brain twister (or if you need a refresher), head back to the original post before checking out the answer. No cheating! ๐Ÿ™‚


How many positive values of x exist if n is a positive integer and 10 < x < 100? n = [x! โ€“ (x โ€“ 2)!]/(x โ€“ 1)! Numeric Entry: __________  

Answer and Explanation

This is a question that can be a total time waster if you donโ€™t proceed logically. In other words, donโ€™t just keep plugging in numbers and seeing if you get an integer: figuring out (20! โ€“ 18!)/19! and the like is a slog.

However, picking up on the pattern allows for a nice Eureka! moment.

See, if you choose a positive value for x, say 5, you get the following:

(5! โ€“ 3!)/4! = [(5)(4)(3)(2)(1) โ€“ (3)(2)(1)]/ (4)(3)(2)(1)

The (3)(2)(1) will cancel out giving you:

[(5)(4) โ€“ 1]/4

Notice that -1.

If you have a number that is divisible by 4, the number 5 x 4, and you subtract by one, you will always get a number that is not divisible by 4.

Of course, the question is asking about larger values but the relationship will hold true:

20! โ€“ 18!/19! = ](20)(19) โ€“ 1]19

Again, you are taking a number (20)(19) that is divisible by the denominator and you are subtracting it by 1, meaning that it wonโ€™t divide evenly. Therefore, the answer is 0.

Hope you enjoyed this challenge! Stay tuned for more.


By the way, students who use Magoosh GRE improve their scores by an average of 8 points on the new scale (150 points on the old scale.) Click here to learn more.

5 Responses to Magoosh Brain Twister: Factorial Fun – Explanation

  1. izz November 7, 2015 at 7:02 pm #

    Why did we leave a 4 in the numerator?

    (5! โ€“ 3!)/4! = [(5)(4)(3)(2)(1) โ€“ (3)(2)(1)]/ (4)(3)(2)(1)

    Isn’t this the same as

    [(5)(4)(3)(2)(1)]/(4)(3)(2)(1) โ€“ [(3)(2)(1)]/ (4)(3)(2)(1)

    Is this is true then we’d be left with

    (5) โ€“ 1/4


    [(5)(4) โ€“ 1]/4

    Both ways we are left with a numerator not divisible by 4 but what am I missing?

  2. shilpa June 28, 2015 at 4:52 am #

    Hey Chris! I love magoosh website…you guys are so helpful!!!!!!

    I’ve already given GRE and TOEFL….I got 304 (154 – Q 150 – V) in gre and 103 in Toefl….I flopped in GRE…universities I am planning for need atleast 160 in quants….I am retaking d exam …..I have all the free materials, your videos, Manhattan 1-8 series , all d materials in world ( .pls do suggest links for more of these…lol ๐Ÿ˜‰ ) I have already finished your vocab builder apps, I keep revising them daily…..the thing is I don’t know how to start, where to start, how to improve ..I need 6 more marks in Q and 2-3 marks in V will definitely be a boost… .I don’t know how to go about it …please suggest me a way .. My current situation doesn’t allow me to enroll in the courses provided in the test prep sites…..some tips on how to go about would do ๐Ÿ™‚

    Thanks in advance! ๐Ÿ™‚

  3. Jo Mama March 28, 2015 at 8:53 am #

    Quite a diabolical problem!

  4. Indumathi March 27, 2015 at 12:04 pm #


    The question asks for how many positive values of x exist between 10 < x < 100, if n is a POSITIVE integer. But the answer 0 is neither positive nor negative. Got confused ๐Ÿ™ Please clarify me.


    • Chris Lele
      Chris Lele March 30, 2015 at 2:44 pm #

      Hi Indu,

      So the question is asking for the total number of values of ‘x’ (not ‘n’) that satisfy the equation. Since there are zero (or no) positive values of ‘x’ that satisfy the equation, the answer is zero.

      Hope that clears things up ๐Ÿ™‚

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