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# What is the smallest integer k for which 7^k > 14*7^15

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What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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18 Nov 2016, 03:04
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What is the smallest integer k for which 7^k > 14*7^15 ?

A. 14
B. 15
C. 16
D. 17
E. 18
[Reveal] Spoiler: OA

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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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18 Nov 2016, 03:23
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Bunuel wrote:
What is the smallest integer k for which 7^k > 14*7^15 ?

A. 14
B. 15
C. 16
D. 17
E. 18

the given can be reduced to
7^k > 2 * 7^16
if 7^k has to be greater then minimum value of k= 17
i.e., 7 * 7^16 > 2 * 7^16
Hence D

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What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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18 Nov 2016, 08:43
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Bunuel wrote:
What is the smallest integer k for which 7^k > 14*7^15 ?

A. 14
B. 15
C. 16
D. 17
E. 18

$$7^k > 14*7^{15}$$

Or, $$7^k > 7*2*7^{15}$$

Or, $$7^k > 2*7^{16}$$

Now, If k = 16

Or, $$7^{16} > 2*7^{16}$$ { Not true since RHS < LHS }

To make LHS > RHS lets make LHS > $$7^{16}$$

If , $$7^{17} > 2*7^{16}$$ Holds good...

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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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18 Nov 2016, 09:59
Bunuel wrote:
What is the smallest integer k for which 7^k > 14*7^15 ?

A. 14
B. 15
C. 16
D. 17
E. 18

14*7^15 = 2* 7^16
since we have 7^k greater than above expression, smallest possible value for K is 17.

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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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18 Nov 2016, 10:52
7^k>14*(7^15)
> 2*(7^16)

If k= 16 it does not satisfy.
If k=17 that is equal to 7*(7^16) which is greater than 2*(7^16)
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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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29 Nov 2016, 15:36
Bunuel wrote:
What is the smallest integer k for which 7^k > 14*7^15 ?

A. 14
B. 15
C. 16
D. 17
E. 18

We are given that 7^k > 14 x 7^15. Let’s first simplify 14 x 7^15:

14 x 7^15 = 2^1 x 7^1 x 7^15 = 2 x 7^16

Thus, 7^k > 2 x 7^16 when k = 17 since 7^17 = 7 x 7^16, which is greater than 2 x 7^16.

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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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20 Dec 2016, 18:42
K= ? Smallest value
$$7^{k} > 14 x 7^{15} 7^{k} > 2 x 7^{1} x 7^{15} 7^{k} > 2 x 7^{16}$$
k = 17
D
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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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22 Jun 2017, 14:50
7^k > 14*7^15 ?

7^k > 7*2*7^15

7^k> 2*7^16

7^16>2*7^16

As RHS is greater, in order to make LHS greater we increase the power.

7^17>2*7^16

So, smallest value of k = 17

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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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25 Dec 2017, 06:27
Bunuel wrote:
What is the smallest integer k for which 7^k > 14*7^15 ?

A. 14
B. 15
C. 16
D. 17
E. 18

$$7^k>14*7^{15}$$
=> $$7^k>2*7*7^{15}$$
=> $$7^k>2*7^{16}$$
=> Therefore k should be > 16.
=> Since smallest integer >16 is 17. Therefore k=17
=> $$7^k=7^{17}=7*7^{16}$$
=> $$7*7^{16}>2*7^{16}$$
=> Therefore $$7^{17}>2*7^{16}$$
=> Therefore k=17

Hence 'D'

Thanks
Dinesh

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Re: What is the smallest integer k for which 7^k > 14*7^15 [#permalink]

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30 Dec 2017, 22:08
Bunuel wrote:
What is the smallest integer k for which 7^k > 14*7^15 ?

A. 14
B. 15
C. 16
D. 17
E. 18

Simplifying we get, 7^k > 2^1 x 7^1 x 7^15

7^k > 2^1 x 7^16

As we can see RHS has 7^16, we need LHS at least 7^17 (7^17 = 7^1 x 7^ 16, which is surely greater than 2 x 7^16)

Therefore k = 17 (D)
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Re: What is the smallest integer k for which 7^k > 14*7^15   [#permalink] 30 Dec 2017, 22:08
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