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

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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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New post 07 Sep 2018, 01:21
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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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Re: If n is the greatest positive integer for which 5^n is a factor of 50!  [#permalink]

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New post 07 Sep 2018, 02:10
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
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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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New post 07 Sep 2018, 02:54
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.
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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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New post 07 Sep 2018, 15:15
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.
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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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New post 09 Sep 2018, 18:28
=>
\(50! = 1*2*…*5*…*10*…*15*…*20*…*25*…*30*…*35*…*40*…*45*…*50\)
\(=1*2*…*(5)…*(2*5)*…*(3*5)*…*(4*5)*…*(52)*…*(6*5)*…*(7*5)*…*(8*5)*…*(9*5)*…*(2*5^2)\)
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
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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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New post 17 Sep 2018, 18:31
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!

Answer: C
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Re: If n is the greatest positive integer for which 5^n is a factor of 50!   [#permalink] 17 Sep 2018, 18:31
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