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Founder  Joined: 18 Apr 2015
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The quantity will end in how many zeros [#permalink]
Expert's post 00:00

Question Stats: 64% (01:57) correct 35% (02:31) wrong based on 74 sessions

The quantity $$3^3 4^4 5^5 6^6$$ - $$3^6 4^5 5^4 6^3$$ will end in how many zeros ?

A. 3

B. 4

C. 5

D. 6

E. 9
[Reveal] Spoiler: OA

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GRE Prep Club Members of the Month: Each member of the month will get three months free access of GRE Prep Club tests. Intern Joined: 08 Dec 2017
Posts: 40
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Re: The quantity will end in how many zeros [#permalink]
1
KUDOS
3^3 4^4 5^5 6^6 - 3^6 4^5 5^4 6^3
=...(2^14)(5^5) - ....(2^13)(5^4)
There should be 4 zeroes ending up this quantity. Director  Joined: 07 Jan 2018
Posts: 694
Followers: 11

Kudos [?]: 724  , given: 88

Re: The quantity will end in how many zeros [#permalink]
3
KUDOS
realize that for a $$0$$ to occur there has to be a multiplication of $$5$$ and $$2$$

simplify the 1st term $$3^3 4^4 5^5 6^6$$= $$3^3* (2^2)^4*5^5* (2*3)^6$$ = $$3^9* 2^1^4* 5^5$$
Similarly simplify the 2nd term that should come out to be $$3^9* 2^1^3* 5^4$$
Subtracting 2nd term from 1st term:take the common term which is whole of the 2nd term

$$3^9*2^1^3*5^4$$$$(10-1)$$
now we have to find out the number of zeros in the common term because non common term is 9
2 and 5 multiply to 10. Here the limiting number is 5 which is equal to 4 hence 4 zeros
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Manager Joined: 15 Feb 2018
Posts: 53
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Kudos [?]: 20 , given: 33

Re: The quantity will end in how many zeros [#permalink]
Hi, I don't quite understand your last sentence. What do you mean by the limiting number?

amorphous wrote:
realize that for a $$0$$ to occur there has to be a multiplication of $$5$$ and $$2$$

simplify the 1st term $$3^3 4^4 5^5 6^6$$= $$3^3* (2^2)^4*5^5* (2*3)^6$$ = $$3^9* 2^1^4* 5^5$$
Similarly simplify the 2nd term that should come out to be $$3^9* 2^1^3* 5^4$$
Subtracting 2nd term from 1st term:take the common term which is whole of the 2nd term

$$3^9*2^1^3*5^4$$$$(10-1)$$
now we have to find out the number of zeros in the common term because non common term is 9
2 and 5 multiply to 10. Here the limiting number is 5 which is equal to 4 hence 4 zeros Director  Joined: 07 Jan 2018
Posts: 694
Followers: 11

Kudos [?]: 724  , given: 88

Re: The quantity will end in how many zeros [#permalink]
2
KUDOS
gremather wrote:
Hi, I don't quite understand your last sentence. What do you mean by the limiting number?

amorphous wrote:
realize that for a $$0$$ to occur there has to be a multiplication of $$5$$ and $$2$$

simplify the 1st term $$3^3 4^4 5^5 6^6$$= $$3^3* (2^2)^4*5^5* (2*3)^6$$ = $$3^9* 2^1^4* 5^5$$
Similarly simplify the 2nd term that should come out to be $$3^9* 2^1^3* 5^4$$
Subtracting 2nd term from 1st term:take the common term which is whole of the 2nd term

$$3^9*2^1^3*5^4$$$$(10-1)$$
now we have to find out the number of zeros in the common term because non common term is 9
2 and 5 multiply to 10. Here the limiting number is 5 which is equal to 4 hence 4 zeros

since for a '$$0$$' to occur $$5$$ has to be multiplied by $$2$$. The number of zeros will depend on the minimum power raised of either of the two numbers

For eg.
$$100 = 5^2 * 2^2 = 2$$ zeros at the end (because both the terms have power raised to 2)
$$125 = 5^3 * 2^0 = 0$$ zeros at the end (because 2 is raised to a power of 0 hence 2 becomes the limiting number)
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Manager Joined: 18 Jun 2019
Posts: 124
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Kudos [?]: 23 , given: 62

Re: The quantity will end in how many zeros [#permalink]
Hi! looking for another solution to this question.

I've simplified upto 3^9*2^13*5^4(9) and don't know how to proceed to fnd the number of zero's. Founder  Joined: 18 Apr 2015
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Re: The quantity will end in how many zeros [#permalink]
1
KUDOS
Expert's post
The trick is how many "five you do have in the quantity ??

we do have nine five numbers.

Now, you will have a zero whenever you do have $$2 \times 5 = 10$$

In our quantity we do have 4 couples of 5 plus one 5 disparaged

5*5
5*5
5*5
5*5
5

That means two couples of five are 4 zeros.

Hope this helps
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Manager Joined: 18 Jun 2019
Posts: 124
Followers: 0

Kudos [?]: 23 , given: 62

Re: The quantity will end in how many zeros [#permalink]
Carcass wrote:
The trick is how many "five you do have in the quantity ??

we do have nine five numbers.

Now, you will have a zero whenever you do have 2 \times 5 = 10

In our quantity we do have 4 couples of 5 plus one 5 disparaged

5*5
5*5
5*5
5*5
5

That means two couples of five are 4 zeros.

Hope this helps

Much clearer!! Thank you!
Senior Manager  Joined: 22 Jun 2019
Posts: 425
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Kudos [?]: 69 , given: 150

Re: The quantity will end in how many zeros [#permalink] Re: The quantity will end in how many zeros   [#permalink] 09 Dec 2019, 02:56
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