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Intern  Status: Yes. It was I who let the dogs out.
Joined: 03 Dec 2012
Posts: 38
H: B
Concentration: General Management, Leadership
GMAT Date: 08-31-2013
If m = 3^n, what is the greatest value of n for which m is a  [#permalink]

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12 00:00

Difficulty:   5% (low)

Question Stats: 79% (01:02) correct 21% (01:35) wrong based on 464 sessions

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If m = 3^n, what is the greatest value of n for which m is a factor of 25!

(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

My Question: Please provide an explanation on how to arrive at the answer.

Disclaimer: I have used the Search Box Before Posting. I used the first sentence of the question or a string of words exactly as they show up in the question below for my search. I did not receive an exact match for my question.

Source: Veritas Prep; Book 02
Chapter: Homework
Topic: Arithmetic
Question: 106
Question: Page 252
Solution: PDF Page 29 of 32

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Originally posted by hb on 23 Jul 2013, 07:52.
Last edited by Bunuel on 23 Jul 2013, 08:00, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 58297
Re: If m = 3^n, what is the greatest value of n for which m is a  [#permalink]

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5
6
If m = 3^n, what is the greatest value of n for which m is a factor of 25!

(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

We should find the highest power of 3 in 25!: $$\frac{25}{3}+\frac{25}{3^2}=8+2=10$$ (take only the quotient into account).. So the highest power of 3 in 25! is 10.

Theory on this: everything-about-factorials-on-the-gmat-85592.html

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Re: If m = 3^n, what is the greatest value of n for which m is a  [#permalink]

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If m=3n, what is the greatest value of n for which m is a factor of 25!?

Solution-
Consider multiples of 25!=> 3,6,9,12,15,18,21,24
Count no. of 3 in each multiple.
3=3x1->1
6=3x2->1
9=3x3->2
12=3x4->1
15=3x5->1
18=3x3x2->2
21=3x7->1
24=3x8->1

---- count 3's =10 so answer is 10

=> 25!/3x3x3x3x3x3x3x3x3x3
Director  S
Joined: 12 Nov 2016
Posts: 701
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Schools: Yale '18
GMAT 1: 650 Q43 V37 GRE 1: Q157 V158 GPA: 2.66
Re: If m = 3^n, what is the greatest value of n for which m is a  [#permalink]

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hb wrote:
If m = 3^n, what is the greatest value of n for which m is a factor of 25!

(A) 8
(B) 10
(C) 12
(D) 14
(E) 16

My Question: Please provide an explanation on how to arrive at the answer.

Disclaimer: I have used the Search Box Before Posting. I used the first sentence of the question or a string of words exactly as they show up in the question below for my search. I did not receive an exact match for my question.

Source: Veritas Prep; Book 02
Chapter: Homework
Topic: Arithmetic
Question: 106
Question: Page 252
Solution: PDF Page 29 of 32

We don't actually have to expand 25 and count all the factors of 3 within the expansion 25- realistically that'll be too inefficient on the GMAT- so instead we can do

25/3 + 25/3^2 = 8 +2 = 10 factors ... but how do we use this for say finding the factors of 4 in "25!"?

25/4 = 6 not minding the remainder just like finding the factors of 0 in a factorial

25/4^2 = 1 again not minding the remainder

6 + 1 =7

why wouldn't we do 25/4^3? because 4^3 =64 and this is too large to go into 25- if we wanted to find the number of 7'2 in 100! it would be the same pattern

100/7 = 14

100/7^2 = 2

14 + 2 = 16 factors of 7 in 100!

Thus
"B"
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Joined: 27 Jul 2017
Posts: 47
Re: If m = 3^n, what is the greatest value of n for which m is a  [#permalink]

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The answer must be B, i.e. 3 will have a total power of 10 in 25!.
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Re: If m = 3^n, what is the greatest value of n for which m is a  [#permalink]

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Let's talk strategy here. Many explanations on this forum focus blindly on the math. While that isn't necessarily bad, remember: the GMAT is a critical-thinking test. While there might be equations you could memorize that give you solutions for problems like this, those equations break down as soon as the GMAT words the question a little bit differently. For those of you studying for the GMAT, you will want to internalize strategies that actually minimize the amount of math that needs to be done, making it easier to manage your time. The tactics I will show you here will be useful for numerous questions, not just this one. My solution is going to walk through not just what the answer is, but how to strategically think about it. Ready? Here is the full "GMAT Jujitsu" for this question:

The Approach

Questions like this are actually designed to test your ability to recognize patterns behind otherwise messy math. $$25!$$ is a big number. Obviously, there is no way you are going to multiply it out. The key is to focus on exactly what the question is asking, looking for logical leverage to strategically attack the question. In this case, the problem asks, "if $$m = 3^n$$, what is the greatest value of $$n$$ for which $$m$$ is a factor of $$25!$$?" In other words, "how many times could we factor a $$3$$ out of $$25$$ factorial?"

Of course, a possible trap would be to just think of the multiples of $$3$$s in the factorial:

$$25*$$ $$(24)$$ $$*23*22*$$ $$(21)$$ $$*20*19*$$ $$(18)$$ $$*17*16*$$ $$(15)$$ $$*14*13*$$ $$(12)$$ $$*11*10*$$ $$(9)$$ $$*8*7*$$ $$(6)$$ $$*5*4*$$ $$(3)$$ $$*2*1$$

There are eight multiples of $$3$$ in $$25!$$, but this is a classic trap (and, incidentally, the most-commonly picked wrong answer.) The way around this bad logic is to realize that the number of "multiples of $$3$$" embedded in a factorial isn't necessarily the same thing as the number of $$3$$s. Some numbers contain more than one factor of $$3$$. For example, $$18 = 2*3*3$$. $$18$$ can contribute two $$3$$s to the total amount. Since the question asks us for the maximum number of $$3$$s we could factor out of $$25!$$, we need to look for all the possible $$3$$s. $$9$$, of course, contains two $$3$$s ($$9=3*3$$). $$18$$ also contains two $$3$$s ($$18=2*3*3$$).
$\begin{matrix} \text{Factor:} & \text{3} & \text{6} & \text{9} & \text{12} & \text{15} & \text{18} & \text{21} & \text{24} \\ \hline \text{# of 3s} & \text{1} & \text{1} & \text{2} & \text{1}& \text{1}& \text{2}& \text{1}& \text{1} & \text{= 10} \end{matrix}$
Thus, $$25!$$ contains $$10$$ total $$3$$s, and the answer is B.

A Hypothetical

Some other answers in this forum have suggested a formula to figure this out: $$\frac{25}{3}+\frac{25}{3^2}=8+2=10$$ (taking only quotients into account.) While this equation technically works for this particular problem, such a formula completely implodes if the problem were to ask, "if $$m = 4^n$$, what is the greatest value of $$n$$...?" The formula would give the solution: $$\frac{25}{4}+\frac{25}{4^2}=6+1=7$$. But this is not even close to the right answer. Since $$4=2*2$$, a factor containing a single $$2$$ can still make a difference, even if it doesn't contain a $$4$$. The number of $$2$$s in this hypothetical question would be:
$\begin{matrix} \text{Factor:} & \text{2} & \text{4} & \text{6} & \text{8} & \text{10} & \text{12} & \text{14} & \text{16} & \text{18} & \text{20} & \text{22} & \text{24} \\ \hline \text{# of 2s} & \text{1} & \text{2} & \text{1} & \text{3}& \text{1}& \text{2}& \text{1}& \text{4}& \text{1}& \text{2}& \text{1}& \text{3} & \text{= 22} \end{matrix}$
With $$22$$ possible $$2$$s embedded in $$25!$$, that would mean we could extract $$11$$ possible $$4$$s... much more than the "equation" calculated.

Takeaways

Now, let’s look back at this problem through the lens of strategy. Your job as you study for the GMAT isn't to memorize the solutions to specific questions; it is to internalize strategic patterns that allow you to solve large numbers of questions. This problem can teach us patterns seen throughout the GMAT. The primary pattern this problem demonstrates is "Mathugliness" -- whereby the GMAT tries to "flex on you," using obnoxious or repetitive math to make the problem seem harder than it actually is. When you encounter Mathugliness, get excited. Look for leverage in the problem that will allow you to solve the problem conceptually, instead of working out all the math. You just need to look at how the problem is structured and determine what would limit or define the value you are looking for. Determine what those limits are, and you have your answer. It is much safer to look for patterns in how factors are distributed, instead of blindly trusting in an equation. Make a table, organize your thoughts, and focus on what the question is asking. The GMAT is notorious for crafting questions in a way that trips up students who blindly rely on "tricks." (Incidentally, can you see the pattern in the table above for the "# of 2s"? Patterns turn "inefficient" math into great critical-thinking opportunities.) And that is thinking like the GMAT.
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Aaron J. Pond
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Hit "+1 Kudos" if my post helped you understand the GMAT better.
Look me up at https://www.veritasprep.com/gmat/aaron-pond/ if you want to learn more GMAT Jujitsu. Re: If m = 3^n, what is the greatest value of n for which m is a   [#permalink] 22 Jan 2019, 17:28
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