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# What is the greatest integer m for which 45^m is a factor of 48!?

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Director
Joined: 18 Feb 2019
Posts: 602
Location: India
GMAT 1: 460 Q42 V13
GPA: 3.6
What is the greatest integer m for which 45^m is a factor of 48!?  [#permalink]

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22 Mar 2019, 11:58
2
00:00

Difficulty:

65% (hard)

Question Stats:

55% (01:55) correct 45% (01:44) wrong based on 29 sessions

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For any integer P greater than 1, P! denotes the product of all the integers from 1 to P, inclusive. What is the greatest integer m for which 45^m is a factor of 48!?

A. 1
B. 2
C. 5
D. 10
E. 11
Math Expert
Joined: 02 Aug 2009
Posts: 8281
Re: What is the greatest integer m for which 45^m is a factor of 48!?  [#permalink]

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22 Mar 2019, 19:47
kiran120680 wrote:
For any integer P greater than 1, P! denotes the product of all the integers from 1 to P, inclusive. What is the greatest integer m for which 45^m is a factor of 48!?

A. 1
B. 2
C. 5
D. 10
E. 11

48! is product of 1 to 48.
Now 45 =$$3^2*5$$, so here we will look at the number of 3^2 or 5, whichever is less, in the product 1*2*...47*48.
Of course, we can clearly say that there are lesser number of 3 than the number of 5, but we may have to check 3^2 vs 5
Number of 5 = $$\frac{48}{5}+\frac{48}{5^2}=9+1=10$$, we take only the integer part before the decimal, so 1.98 will be 1.
Number of 3 = $$\frac{48}{3}+\frac{48}{3^2}+\frac{48}{3^3}=16+5+1=22$$, so number of 3^2 will be 22/2=11

Now, 10 is lesser than 11, so our answer will be 10

D
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Re: What is the greatest integer m for which 45^m is a factor of 48!?   [#permalink] 22 Mar 2019, 19:47
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