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# Find the sum of the digit of the least number K

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Intern
Joined: 01 May 2017
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Location: United States (IL)
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Find the sum of the digit of the least number K  [#permalink]

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14 Mar 2018, 04:40
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Difficulty:

55% (hard)

Question Stats:

61% (02:42) correct 39% (01:46) wrong based on 54 sessions

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Find the sum of the digit of the least number K such that 2K is a square and 3K is a cube.

1. 9
2. 8
3. 13
4. 15
6. 16
Senior PS Moderator
Joined: 26 Feb 2016
Posts: 3334
Location: India
GPA: 3.12
Find the sum of the digit of the least number K  [#permalink]

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14 Mar 2018, 05:27
1
1
Find the sum of the digit of the least number K such that 2K is a square and 3K is a cube.

1. 9
2. 8
3. 13
4. 15
6. 16

The proportion of the number, a square, and a cube is 1:2:3

If the number is $$3^2*2^3$$, $$3^2*2^3$$$$*2$$ is the square and $$3^2*2^3$$$$*3$$ is the cube

Therefore, the number will be 72($$3^2*2^3$$) and the sum of digits is 9(Option A)

P.S In this problem, the number would have one 2 lesser than a perfect square
and one 3 lesser than a perfect cube. That's the reason for the choice of number $$2^3*3^2$$
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Find the sum of the digit of the least number K  [#permalink]

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14 Mar 2018, 11:29
pushpitkc wrote:
Find the sum of the digit of the least number K such that 2K is a square and 3K is a cube.

1. 9
2. 8
3. 13
4. 15
6. 16

The proportion of the number, a square, and a cube is 1:2:3

If the number is $$3^2*2^3$$, $$3^2*2^3$$$$*2$$ is the square and $$3^2*2^3$$$$*3$$ is the cube

Therefore, the number will be 72($$3^2*2^3$$) and the sum of digits is 9(Option A)

Hi pushpitkc,

Can you please elaborate your explanation. I did not quite understand where The proportion of the number, a square, and a cube is 1:2:3 came from?

Thanks.
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Re: Find the sum of the digit of the least number K  [#permalink]

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14 Mar 2018, 11:36
Hi rohan2345

The question stem explicitly states that for the number K - 2K is a square and 3K is a cube.
Therefore, the proportion of the number K, its square, and its cube is K:2K:3K (1:2:3)

Hope that helps!
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Find the sum of the digit of the least number K  [#permalink]

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Updated on: 17 Mar 2018, 08:58
1
Find the sum of the digit of the least number K such that 2K is a square and 3K is a cube.

1. 9
2. 8
3. 13
4. 15
6. 16

if 2K is a square,
then 2K=2*2*n*n
thus K=2*n*n
and 3K=3*2*n*n
if 3K is a cube,
then n=3*2=6
thus K=2*6*6=72
7+2=9

Originally posted by gracie on 14 Mar 2018, 19:50.
Last edited by gracie on 17 Mar 2018, 08:58, edited 1 time in total.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8810
Location: Pune, India
Re: Find the sum of the digit of the least number K  [#permalink]

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16 Mar 2018, 04:39
3
Find the sum of the digit of the least number K such that 2K is a square and 3K is a cube.

1. 9
2. 8
3. 13
4. 15
6. 16

Take it one step at a time.

2k is a square. A square needs to have even powers of every prime factor. So k must already have one or three or five or ... 2s.

The smallest acceptable value of k would be 2. k could also be $$2^3$$ or $$2^5$$ etc.

2*k = 2*2 is a perfect square. If k has another prime factor, it must ALREADY exist in even power.

3K is a cube. A cube needs to have all powers of prime factors as multiples of 3.
So k must have two or five or eight ... 3s.

The smallest acceptable value of k would be 3*3

3*k = 3*3*3 is a perfect cube. Any other prime factor it may have must ALREADY be in power of 3. So the power of 2 in k must be 3.

So we see that k must have at least three 2s and two 3s.
$$k = 2^3 * 3^2 = 72$$
Sum of the digits = 7+2 = 9
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Re: Find the sum of the digit of the least number K  [#permalink]

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17 Mar 2018, 01:31
Can anyone post the link to the topic this question is covering ?
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Location: United States (IL)
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Re: Find the sum of the digit of the least number K  [#permalink]

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17 Mar 2018, 06:44
rohail786911 wrote:
Can anyone post the link to the topic this question is covering ?

You may refer to this - https://gmatclub.com/forum/divisibility ... 74998.html
Re: Find the sum of the digit of the least number K &nbs [#permalink] 17 Mar 2018, 06:44
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