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If n is the greatest positive integer for which 5^n is a factor of 50!

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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If n is the greatest positive integer for which 5^n is a factor of 50!  [#permalink]

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07 Sep 2018, 00:21
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[Math Revolution GMAT math practice question]

If $$n$$ is the greatest positive integer for which $$5^n$$ is a factor of $$50!$$, what is the value of $$n$$?

$$A. 10$$
$$B. 11$$
$$C. 12$$
$$D. 13$$
$$E. 14$$

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The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Director Joined: 06 Jan 2015 Posts: 515 Location: India Concentration: Operations, Finance GPA: 3.35 WE: Information Technology (Computer Software) Re: If n is the greatest positive integer for which 5^n is a factor of 50! [#permalink] Show Tags 07 Sep 2018, 01:10 MathRevolution wrote: [Math Revolution GMAT math practice question] If $$n$$ is the greatest positive integer for which $$5^n$$ is a factor of $$50!$$, what is the value of $$n$$? $$A. 10$$ $$B. 11$$ $$C. 12$$ $$D. 13$$ $$E. 14$$ $$50!$$/$$5^n$$ ==> 10+2=12 Hence C _________________ आत्मनॊ मोक्षार्थम् जगद्धिताय च Resource: GMATPrep RCs With Solution Director Joined: 31 Oct 2013 Posts: 786 Concentration: Accounting, Finance GPA: 3.68 WE: Analyst (Accounting) Re: If n is the greatest positive integer for which 5^n is a factor of 50! [#permalink] Show Tags 07 Sep 2018, 01:54 1 MathRevolution wrote: [Math Revolution GMAT math practice question] If $$n$$ is the greatest positive integer for which $$5^n$$ is a factor of $$50!$$, what is the value of $$n$$? $$A. 10$$ $$B. 11$$ $$C. 12$$ $$D. 13$$ $$E. 14$$ we are looking for the number of n in 50! as a factor. Note: n is the greatest positive integer. The way through which we can determine the numbers of n in 50!: 50/5 = 10 50/25 = 2 So the greatest number of n in 50! is 12. The best answer is C. GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 477 Re: If n is the greatest positive integer for which 5^n is a factor of 50! [#permalink] Show Tags 07 Sep 2018, 14:15 MathRevolution wrote: [Math Revolution GMAT math practice question] If $$n$$ is the greatest positive integer for which $$5^n$$ is a factor of $$50!$$, what is the value of $$n$$? $$A. 10$$ $$B. 11$$ $$C. 12$$ $$D. 13$$ $$E. 14$$ $$n \geqslant \,\,1\,\,\,\operatorname{int}$$ $$\frac{{50!}}{{{5^n}}} = \operatorname{int}$$ $$?\,\, = \,\,\max \,\,n$$ In English: how many primes equal to 5 are we able to find in the product 50*49*48*47*...*6*5*4*3*2*1 ? -------------------------------------------------------------------------------------------------------------- In 5 = 5*1 we find the first In 10 = 5*2 we find the second In 15 = 5*3 we find the third In 20 = 5*4 we find the four In 25 = 5*5 we find the fifth and the sixth, but forget the sixth for a moment, please! In 30 = 5*6 we find the "sixth" (yes, that´s a lie... wait a bit!) In 35 = 5*7 we find the "seventh" (wait...) --- etc --- In 50 = 5*10 we find the "tenth" (second mistake... because 5*10 = 5*5*2 .... but wait...) -------------------------------------------------------------------------------------------------------------- What is going on here? When we calculate 50/5 = 10 , we find the "first" 5´s involved... with some lies... When we divide 50 by 5^2 , we find exactly the numbers like 25 (and 50) , in which there are (at least) two 5´s in it, and only one 5 (in each case) was counted previously (between the parallel lines) ... Now they were properly counted... no more lies! Conclusion: 50/5 + 50/25 = 12 is the right answer! (In this case, there are no 5^3 , 5^4 , ... in 50! . In other words, each of the integers 1, 2, 3, 4, 5, ... , 50 has at most two 5´s in its corresponding prime decomposition!) Now the "recipe" (used correctly in previous posts): $$? = \left\lfloor {\frac{{50}}{5}} \right\rfloor + \left\lfloor {\frac{{50}}{{{5^2}}}} \right\rfloor + \left\lfloor {\frac{{50}}{{{5^3}}}} \right\rfloor + \ldots = 10 + 2 + 0 + 0 + 0 + \ldots = \boxed{12}$$ where $$\left\lfloor N \right\rfloor$$ denotes the "floor" of N, that is, the greatest integer that is less than, or equal to, N. If you prefer: when we divide 50 by 5, we have quotient 10 (the floor) and the remainder 0 (irrelevant when looking for the floor)! Another example: GMATH wrote: If n is the greatest positive integer for which 3^n is a factor of 50! , what is the value of n? A. 19 B. 20 C. 21 D. 22 E. 23 Answer: $$? = \left\lfloor {\frac{{50}}{3}} \right\rfloor + \left\lfloor {\frac{{50}}{{{3^2}}}} \right\rfloor + \left\lfloor {\frac{{50}}{{{3^3}}}} \right\rfloor + \left\lfloor {\frac{{50}}{{{3^4}}}} \right\rfloor + \ldots = 16 + 5 + 1 + 0 + \ldots = \boxed{22}$$ Fabio, do you think it is useful to remember this "recipe" for GMAT purposes? YES, although it´s obviously unprobable you will need it. (This is MUCH less important than to know that the [length of the] height of the equilateral triangle is the [length of the] side times half the square root of 3, for instance.) But... If you understood the recipe, it will be much easier to remember it (and to apply it) correctly... That´s the GMATH´s method "essence": REAL AND DEEP UNDERSTANDING! Regards, fskilnik. _________________ Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6517 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If n is the greatest positive integer for which 5^n is a factor of 50! [#permalink] Show Tags 09 Sep 2018, 17:28 => $$50! = 1*2*…*5*…*10*…*15*…*20*…*25*…*30*…*35*…*40*…*45*…*50$$ $$=1*2*…*(5)…*(2*5)*…*(3*5)*…*(4*5)*…*(52)*…*(6*5)*…*(7*5)*…*(8*5)*…*(9*5)*…*(2*5^2)$$ Since $$5$$ is a prime number, no further factors of $$5$$ appear in the prime factorization of $$50!$$. The number of $$5s$$ in the above expansion of $$50!$$ is $$12$$. Therefore, the answer is C. Answer: C _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Re: If n is the greatest positive integer for which 5^n is a factor of 50!  [#permalink]

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17 Sep 2018, 17:31
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If $$n$$ is the greatest positive integer for which $$5^n$$ is a factor of $$50!$$, what is the value of $$n$$?

$$A. 10$$
$$B. 11$$
$$C. 12$$
$$D. 13$$
$$E. 14$$

To determine the number of 5s within 50!, we can use the following shortcut in which we divide 50 by 5, then divide the quotient of 50/5 by 5 and continue this process until we no longer get a nonzero quotient.

50/5 = 10

10/5 = 2

Since 2/5 does not produce a nonzero quotient, we can stop.

The final step is to add up our quotients; that sum represents the number of factors of 5 within 50!.

Thus, there are 10 + 2 = 12 factors of 5 within 50!

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Re: If n is the greatest positive integer for which 5^n is a factor of 50! &nbs [#permalink] 17 Sep 2018, 17:31
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If n is the greatest positive integer for which 5^n is a factor of 50!

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