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# If x, y, and k are positive integers, is k < 10 ?

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If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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Updated on: 14 Oct 2018, 17:10
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Difficulty:

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Question Stats:

24% (02:07) correct 76% (01:51) wrong based on 159 sessions

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If x, y, and k are positive integers, is k < 10 ?

(1) $$45! = x(10^k)$$

(2) $$y=\sqrt[3]{1.25*10^k}$$

Originally posted by harish1986 on 08 Oct 2018, 07:53.
Last edited by chetan2u on 14 Oct 2018, 17:10, edited 3 times in total.
Renamed the topic, corrected the OA
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Joined: 02 Aug 2009
Posts: 7038
Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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08 Oct 2018, 08:05
4
1
If x, y, and k are positive integers, is k < 10 ?

1. $$45! = x(10^k)$$
Max possible value of k is when x does not have any 10s in it..
So number of 10s in 45! is number of 5s = $$\frac{45}{5}+\frac{45}{25}=9+1=10$$
So if x does not have any 5s then k is 10, otherwise it will be <10
Insufficient

2. y=$$3\sqrt{1.25(10^k)}$$
I believe it is the cube root ..
So $$y^3=1.25*10^k=5^3*10^{k-2}$$
So minimum value of k is 2 or k can be 5,8,11 and so on
Insufficient

Combined k cannot be 10 as per statement II and $$k\leq{10}$$
So k<10
Sufficient

C

Editing the OA as it cannot be A
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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14 Oct 2018, 10:57
Hi chetan2u,
I took the statement 2 as 3*(1.25(10k))^-2
and hence i got E. Please confirm if my understanding is incorrect or question is written in a bad format.
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Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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14 Oct 2018, 13:02
harish1986 wrote:
If x, y, and k are positive integers, is k < 10 ?

(1) $$45! = x(10^k)$$

(2) $$y$$ is the cubic root of $$1.25*(10^k)$$

$$x,y,k\,\, \ge 1\,\,\,{\rm{ints}}\,\,\,\left( * \right)$$

$$k\,\,\mathop < \limits^? \,\,10$$

$$\left( 1 \right)\,\,45! = x \cdot {10^k}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,k\,\,\, \le \,\,\,\left\lfloor {{{45} \over 5}} \right\rfloor + \left\lfloor {{{45} \over {25}}} \right\rfloor \, = 10\,\,\,\,\,\left( {**} \right)\,\,$$

$$\left( {**} \right)$$ See my explanation (and notation) here:
https://gmatclub.com/forum/if-n-is-the- ... 75460.html

$$\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {k{\kern 1pt} \,;\,x} \right) = \left( {10\,;\,\,{{45!} \over {{{10}^{10}}}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\,\,\left( {x = {{45!} \over {{{10}^{10}}}}\,\, \ge 1\,\,{\mathop{\rm int}} \,\,\,{\rm{by}}\,\,\left( {**} \right)} \right)\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {k\,;\,x} \right) = \left( {1\,;\,\,{{45!} \over {10}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\,y = \root {3\,} \of {{5 \over 4}\left( {{{10}^k}} \right)} \,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,{5 \over 4}\left( {{2^k} \cdot {5^k}} \right) = {2^{k - 2}} \cdot {5^{k + 1}}\,\,\,\,{\rm{positive}}\,\,{\rm{perfect}}\,\,{\rm{cube}}\,\,\,\,\,$$

$$\Rightarrow \,\,\,\,\,\,\left\{ \matrix{ \,k - 2 = {\rm{mult}}\,\,{\rm{of}}\,\,{\rm{3}} \hfill \cr k + 1 = \,\,{\rm{mult}}\,\,{\rm{of}}\,\,3 \hfill \cr} \right.\,\,\,\,\,\,\,\,\left( {k \ge 2} \right)\,\,\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,k \ge 2\,\,\,\,\,{\rm{divided}}\,\,{\rm{by}}\,\,3\,\,\,{\rm{has}}\,\,{\rm{remainder}}\,\,2\,$$

$$\left\{ \matrix{ \,{\rm{Take}}\,\,k = 2\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,k = 2 + 3 \cdot 3\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,\,\,\left\{ \matrix{ \,k \le 10\,\,\,\,{\rm{by}}\,\,\,\,\left( 1 \right) \cap \left( {**} \right) \hfill \cr \,k \ne 10\,\,\,{\rm{by}}\,\,\left( 2 \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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14 Oct 2018, 13:18
P.S.: $$y = 3 \cdot \,\,\root {} \of {{5 \over 4}\left( {{{10}^k}} \right)}$$ cannot be an integer, for any given positive integer $$k$$.

Reason:

$$y = \,\,\sqrt {\,9 \cdot {5 \over 4}\left( {{2^k} \cdot {5^k}} \right)} \,\,\,\, = \,\,\,\,\sqrt {\,{2^{k - 2}} \cdot {3^2} \cdot {5^{k + 1}}} \,\,\,\,\,\,\left( {k \ge 1\,\,\,{\mathop{\rm int}} } \right)\,\,\,$$

$${2^{k - 2}} \cdot {3^2} \cdot {5^{k + 1}}\,\,\,{\rm{perfect}}\,\,{\rm{square}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\{ \matrix{ \,k - 2\,\,\, \ge 0\,\,\,{\rm{even}} \hfill \cr \,k + 1\,\,\, \ge 0\,\,\,{\rm{even}} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{impossible}}!\,$$

Regards,
Fabio.
_________________

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Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

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Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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14 Oct 2018, 14:17
chetan2u wrote:
If x, y, and k are positive integers, is k < 10 ?

1. $$45! = x(10^k)$$
Max possible value of k is when x does not have any 10s in it..
So number of 10s in 45! is number of 5s = $$\frac{45}{5}+\frac{45}{25}=9+1=10$$
So if x does not have any 5s then k is 10, otherwise it will be <10
Insufficient

2. y=$$3\sqrt{1.25(10^k)}$$
I believe it is the cube root ..
So $$y^3=1.25*10^k=5^3*10^{k-2}$$
So minimum value of k is 2 or k can be 5,8,11 and so on
Insufficient

Combined k cannot be 10 as per statement II and $$k\leq{10}$$
So k<10
Sufficient

C

Editing the OA as it cannot be A

Can you please re-format statement 2? I does not show cubic root.

Thanks
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Joined: 02 Aug 2009
Posts: 7038
Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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14 Oct 2018, 17:11
asthagupta wrote:
Hi chetan2u,
I took the statement 2 as 3*(1.25(10k))^-2
and hence i got E. Please confirm if my understanding is incorrect or question is written in a bad format.

Yes, you are correct.
It is bad formatting, which I have corrected now.
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Manager
Joined: 04 Oct 2017
Posts: 67
Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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14 Oct 2018, 18:14
[quote="chetan2u"]If x, y, and k are positive integers, is k < 10 ?

1. $$45! = x(10^k)$$
Max possible value of k is when x does not have any 10s in it..
So number of 10s in 45! is number of 5s = $$\frac{45}{5}+\frac{45}{25}=9+1=10$$
So if x does not have any 5s then k is 10, otherwise it will be <10
Insufficient

2. y=$$3\sqrt{1.25(10^k)}$$
I believe it is the cube root ..
So $$y^3=1.25*10^k=5^3*10^{k-2}$$
So minimum value of k is 2 or k can be 5,8,11 and so on
Insufficient

Combined k cannot be 10 as per statement II and [m]k\leq{10}[

Hi,

Please explain Number of 10s in 45! is number of 5s. I didnot understand this. Am lost with factorials. I understood the second statement.

Thanks.
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Joined: 21 Jun 2017
Posts: 140
Concentration: Finance, Economics
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If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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14 Oct 2018, 20:24
1
Hi Kezia9

I'll try to explain your query.

In order to find number of zeros or 5 in a complex no , following has to be done.

1. list down numbers ending with 5 and 0
With our case in hand , the list will be
5 10 15 20 25 30 35 40 and 45
2. each of these will have one 5 in them.
3. watchout for perfect squares ...25 will have two.

so 1+1+1+1+2+1+1+1+1= 10 5's

PS: Always remember in order to find zeros or number of 5's you don't need to calculate 2's, because they are plenty. Every alternative number will be even hence, we will just find out the limiting 5.

Posted from my mobile device
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If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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15 Oct 2018, 05:45
1
Kezia9 wrote:
Please explain Number of 10s in 45! is number of 5s. I didnot understand this. Am lost with factorials.
Thanks.

Hi, Kezia9!

I have explained this (step-by-step) here:
https://gmatclub.com/forum/if-n-is-the- ... 75460.html

Regards,
Fabio.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Manager
Joined: 04 Oct 2017
Posts: 67
Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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15 Oct 2018, 08:36
ShankSouljaBoi wrote:
Hi Kezia9

I'll try to explain your query.

In order to find number of zeros or 5 in a complex no , following has to be done.

1. list down numbers ending with 5 and 0
With our case in hand , the list will be
5 10 15 20 25 30 35 40 and 45
2. each of these will have one 5 in them.
3. watchout for perfect squares ...25 will have two.

so 1+1+1+1+2+1+1+1+1= 10 5's

PS: Always remember in order to find zeros or number of 5's you don't need to calculate 2's, because they are plenty. Every alternative number will be even hence, we will just find out the limiting 5.

Posted from my mobile device

Appreciate your efforts to clarify my doubts. THANK YOU.
Manager
Joined: 04 Oct 2017
Posts: 67
Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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15 Oct 2018, 08:37
1
fskilnik wrote:
Kezia9 wrote:
Please explain Number of 10s in 45! is number of 5s. I didnot understand this. Am lost with factorials.
Thanks.

Hi, Kezia9!

I have explained this (step-by-step) here:
https://gmatclub.com/forum/if-n-is-the- ... 75460.html

Regards,
Fabio.

Appreciate your efforts to clarify my doubts. THANK YOU.
Intern
Joined: 27 Sep 2017
Posts: 3
Re: If x, y, and k are positive integers, is k < 10 ?  [#permalink]

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19 Oct 2018, 22:32
chetan2u wrote:
If x, y, and k are positive integers, is k < 10 ?

1. $$45! = x(10^k)$$
Max possible value of k is when x does not have any 10s in it..
So number of 10s in 45! is number of 5s = $$\frac{45}{5}+\frac{45}{25}=9+1=10$$
So if x does not have any 5s then k is 10, otherwise it will be <10
Insufficient

2. y=$$3\sqrt{1.25(10^k)}$$
I believe it is the cube root ..
So $$y^3=1.25*10^k=5^3*10^{k-2}$$
So minimum value of k is 2 or k can be 5,8,11 and so on
Insufficient

Combined k cannot be 10 as per statement II and $$k\leq{10}$$
So k<10

Sufficient

C

Editing the OA as it cannot be A

how the values of k can be determined??//
Re: If x, y, and k are positive integers, is k < 10 ? &nbs [#permalink] 19 Oct 2018, 22:32
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