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# largest possible values

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Intern
Joined: 21 Feb 2010
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14 May 2010, 03:56
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If m is a positive integer and 75^3 is a multiple of 5^m , what is the largest possible values for m?

3
4
5
6
7

Can somebody explain how to solve it?
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14 May 2010, 06:54
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$$75^3 = 75*75*75=(15*5)*(15*5)*(15*5)=(3*5*5)*(3*5*5)*(3*5*5)=3^3*5^6=27*5^6$$

if $$27*5^6$$ is a multiple of $$5^m$$, then $$\frac{27*5^6}{5^m}$$ should result in an integer. Also, remember that m is an integer.

Let us look at the answer options.

A. if m = 3, then the expression becomes $$\frac{27*5^6}{5^3}$$, which leaves $$27*5^3$$, an integer.
B. if m = 4, then the expression becomes $$\frac{27*5^6}{5^4}$$, which leaves $$27*5^2$$, an integer.
C. if m = 5, then the expression becomes $$\frac{27*5^6}{5^5}$$, which leaves $$27*5^1$$, an integer.
D. if m = 6, then the expression becomes $$\frac{27*5^6}{5^6}$$, which leaves $$27*5^0$$, an integer.
E. if m = 7, then the expression becomes $$\frac{27*5^6}{5^7}$$, which leaves $$\frac{27}{5}$$, NOT an integer.

So, out of all the 4 choices that leave us with an integer result, the largest value of m is 6. Hence correct answer is D.
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14 May 2010, 06:17
IMO m=6. here is why

75^3=(25X3)^3=(5^2 X 3)^3 = 5^6 x3^3

which means that largest factor would be 5^6

What is the OA ?
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14 May 2010, 08:30
great explanation agreed wid hideyoshi oa shud be d
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16 May 2010, 08:22
Hi,
I think the simplest method is...

Find out the highest factor of 5s in 75^3. i.e., 5^6 so, m=6. Because that will be the HCF.
Re: largest possible values   [#permalink] 16 May 2010, 08:22
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