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• ### $450 Tuition Credit & Official CAT Packs FREE November 15, 2018 November 15, 2018 10:00 PM MST 11:00 PM MST EMPOWERgmat is giving away the complete Official GMAT Exam Pack collection worth$100 with the 3 Month Pack ($299) • ### Free GMAT Strategy Webinar November 17, 2018 November 17, 2018 07:00 AM PST 09:00 AM PST Nov. 17, 7 AM PST. Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT. # Let n~ be defined for all positive integers n as the remainder when (n  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics Author Message TAGS: ### Hide Tags Current Student Joined: 12 Aug 2015 Posts: 287 Concentration: General Management, Operations GMAT 1: 640 Q40 V37 GMAT 2: 650 Q43 V36 GMAT 3: 600 Q47 V27 GPA: 3.3 WE: Management Consulting (Consulting) Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 05 Nov 2015, 04:57 4 15 00:00 Difficulty: 25% (medium) Question Stats: 70% (01:21) correct 30% (01:46) wrong based on 455 sessions ### HideShow timer Statistics Let n~ be defined for all positive integers n as the remainder when (n - 1)! is divided by n. What is the value of 32~ ? A. 0 B. 1 C. 2 D. 8 E. 31 _________________ KUDO me plenty Intern Joined: 21 Aug 2013 Posts: 3 Concentration: International Business, Leadership GPA: 2.9 WE: Programming (Computer Software) Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 05 Nov 2015, 06:14 2 2 n~ = (n-1)! so 32~ = (32-1)! = 31! when 31!/32 we have 16*2 inside 31! hence 32 gets cancelled and we get remainder as 0 Current Student Joined: 12 Aug 2015 Posts: 2633 Schools: Boston U '20 (M) GRE 1: Q169 V154 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 16 Mar 2016, 00:57 2 Director Joined: 12 Nov 2016 Posts: 743 Location: United States Schools: Yale '18 GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 15 Apr 2017, 12:39 1 Ok so I was initially not able to do this problem because I didn't understand how to divide 31!/32... Now that you guys have made the problem more clear I have a question- 5!/15 must be an integer? Because 5x4x3x2x1 contains the factors of 15 (5 x 3) When I do 17!/30 in a calculator the result is not an integer? Math Expert Joined: 02 Sep 2009 Posts: 50578 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 16 Apr 2017, 00:27 1 Nunuboy1994 wrote: Ok so I was initially not able to do this problem because I didn't understand how to divide 31!/32... Now that you guys have made the problem more clear I have a question- 5!/15 must be an integer? Because 5x4x3x2x1 contains the factors of 15 (5 x 3) When I do 17!/30 in a calculator the result is not an integer? First of all, 17!/30 = 11856247603200 = integer, but what 17!/30 has to do with this problem? _________________ Director Joined: 12 Nov 2016 Posts: 743 Location: United States Schools: Yale '18 GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 16 Apr 2017, 18:09 1 Bunuel wrote: Nunuboy1994 wrote: Ok so I was initially not able to do this problem because I didn't understand how to divide 31!/32... Now that you guys have made the problem more clear I have a question- 5!/15 must be an integer? Because 5x4x3x2x1 contains the factors of 15 (5 x 3) When I do 17!/30 in a calculator the result is not an integer? First of all, 17!/30 = 11856247603200 = integer, but what 17!/30 has to do with this problem? I trying to demonstrate the concept in this problem, which is new to me, in a different example- I think the calculator's notation just makes it appear as 1.something^e even though that doesn't necessarily mean it's not an integer. It's clear now. Math Expert Joined: 02 Sep 2009 Posts: 50578 Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 16 Apr 2017, 22:51 1 Nunuboy1994 wrote: Bunuel wrote: Nunuboy1994 wrote: Ok so I was initially not able to do this problem because I didn't understand how to divide 31!/32... Now that you guys have made the problem more clear I have a question- 5!/15 must be an integer? Because 5x4x3x2x1 contains the factors of 15 (5 x 3) When I do 17!/30 in a calculator the result is not an integer? First of all, 17!/30 = 11856247603200 = integer, but what 17!/30 has to do with this problem? I trying to demonstrate the concept in this problem, which is new to me, in a different example- I think the calculator's notation just makes it appear as 1.something^e even though that doesn't necessarily mean it's not an integer. It's clear now. For big numbers do not user calculator (unless it's not advanced) or Excell, use Wolframalpha: https://www.wolframalpha.com/ Hope it helps. _________________ Director Joined: 12 Nov 2016 Posts: 743 Location: United States Schools: Yale '18 GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 27 Jun 2017, 15:09 shasadou wrote: Let n~ be defined for all positive integers n as the remainder when (n - 1)! is divided by n. What is the value of 32~ ? A. 0 B. 1 C. 2 D. 8 E. 31 Simple- all this question is asking is if you have 31!/32 then what is the remainder? You don't necessarily have to expand 31!- 32 fits in because you have 16 and 2 so there is no remainder Thus "A" Retired Moderator Joined: 19 Mar 2014 Posts: 945 Location: India Concentration: Finance, Entrepreneurship GPA: 3.5 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 09 Jul 2017, 03:12 shasadou wrote: Let n~ be defined for all positive integers n as the remainder when (n - 1)! is divided by n. What is the value of 32~ ? A. 0 B. 1 C. 2 D. 8 E. 31 $$= \frac{(32-1)!}{32}$$ $$= \frac{31!}{32}$$ Numbers $$8 * 4 = 32$$, will cancel out the denomintor and hence the reminder will be ZERO. Hence, Answer is A _________________ "Nothing in this world can take the place of persistence. Talent will not: nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not: the world is full of educated derelicts. Persistence and determination alone are omnipotent." Best AWA Template: https://gmatclub.com/forum/how-to-get-6-0-awa-my-guide-64327.html#p470475 Study Buddy Forum Moderator Joined: 04 Sep 2016 Posts: 1247 Location: India WE: Engineering (Other) Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 10 Nov 2017, 03:20 Bunuel VeritasPrepKarishma 32! has various multiples of 32 built in say 8 * 4, 16* 2. Will not various no of multiples affect my remainder? For eg, even if 8*4 will cancel out 32 in denominator leaving remainder = 0, I still have one multiple of 16*2, correct? Target Test Prep Representative Status: Head GMAT Instructor Affiliations: Target Test Prep Joined: 04 Mar 2011 Posts: 2830 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 14 Nov 2017, 06:08 shasadou wrote: Let n~ be defined for all positive integers n as the remainder when (n - 1)! is divided by n. What is the value of 32~ ? A. 0 B. 1 C. 2 D. 8 E. 31 32~ = (32 - 1)!/32 = 31!/32 = 31!/2^5 Since we can safely say that there are at least five 2s in 31! (for example, 31! has the factors 16 = 2^4 and 8=2^3), the remainder is zero. Alternate Solution: 32~ = (32 - 1)!/32 = 31!/32 = 31!/2^5 We want to know the remainder when 31! is divided by 2^5. If we can establish that there are at least 5 factors of 2 in 31!, then we will know that 2^5 evenly divides into 31!, which means that the remainder would be 0. Let’s determine if we can find at least 5 twos in 31!: 31! = 31 x 30 x 29 x 28 x 27 x 26 x 25 x 24 x … x 1 31! = 31 x (2 x 15) x 29 x (2 X 2 x 14) x 27 x (2 x 13) x 25 x (2 x 2 x 2 x 3) x … x 1 Notice that we have found seven 2s, which is two more than what we needed. Thus, we know that 2^5 will evenly divide into 31!, leaving a remainder of 0. Answer: A _________________ Jeffery Miller Head of GMAT Instruction GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions Manager Joined: 18 Jun 2017 Posts: 50 GMAT 1: 660 Q39 V40 GMAT 2: 700 Q45 V41 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 30 Nov 2017, 22:24 Hi, So basically as long as there are enough prime factor powers in the numerator to cancel out all the prime factor powers in the denominator, the remainder is 0 correct? Math Expert Joined: 02 Sep 2009 Posts: 50578 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 30 Nov 2017, 23:04 calappa1234 wrote: Hi, So basically as long as there are enough prime factor powers in the numerator to cancel out all the prime factor powers in the denominator, the remainder is 0 correct? Yes. If the numerator contains the same (or more) primes as denominator and the (positive integer) powers of primes in the numerator are at least as big as the powers of the same primes in the denominator, then the result of the division will be an integer, so the remainder will be 0. _________________ EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 12853 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: Let n~ be defined for all positive integers n as the remainder when (n [#permalink] ### Show Tags 06 Feb 2018, 11:12 Hi All, This prompt is an example of a 'Symbolism' question - the prompt 'makes up' a math symbol, tells you what it means and asks you to perform a calculation using it. Based on the information in the prompt, we're told that.... 32~ is the remainder when (32-1)! is divided by 32. Now, there's no way that the GMAT would expect you to calculate the value of 31!, so we have to think in terms of what 31! actually is. Here's an example that's a bit easier to deal with: 4! = (4)(3)(2)(1) = 24 What numbers divide EVENLY into 24? 1, 2, 3, 4, 6, 8, 12 and 24 You can clearly see why 1, 2, 3 and 4 divide in - they're in the 'chain' of numbers that are multiplied together. 6 divides in because (2)(3) = 6 - and you can see the (2) and the (3) in the 'chain' Similarly, 8, 12 and 24 are also 'combinations' of the numbers in the 'chain', so they divide evenly in too. 31! has LOTS of numbers in it, so it's evenly divisibly by LOTS of different integers. If you were to write out 31!, you would see a (2) and a (16). This means that (2)(16) = 32 divides evenly into 31!, so there will be a remainder of 0. Final Answer: GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Let n~ be defined for all positive integers n as the remainder when (n  [#permalink]

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07 Feb 2018, 08:29
Bunuel wrote:
calappa1234 wrote:
Hi,

So basically as long as there are enough prime factor powers in the numerator to cancel out all the prime factor powers in the denominator, the remainder is 0 correct?

Yes. If the numerator contains the same (or more) primes as denominator and the (positive integer) powers of primes in the numerator are at least as big as the powers of the same primes in the denominator, then the result of the division will be an integer, so the remainder will be 0.

Hi Bunuel, long time no hear no see

Here is my solution.

to define if $$31!$$ is divisible by $$32$$ make prime factorization of $$32$$ so we get this --- > $$2^4$$

$$31!$$ certainly will have four $$2$$ as prime factors, hence $$\frac{31!}{32}$$ we get integer.

if this solution correct. What if similar question would have remainder - how could i find a remainder ?

i would appreciate your explanation

thank you
Let n~ be defined for all positive integers n as the remainder when (n &nbs [#permalink] 07 Feb 2018, 08:29
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# Let n~ be defined for all positive integers n as the remainder when (n

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