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Find the sum of the digit of the least number K

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Find the sum of the digit of the least number K  [#permalink]

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New post 14 Mar 2018, 05:40
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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
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Find the sum of the digit of the least number K  [#permalink]

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New post 14 Mar 2018, 06:27
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prady2231 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)


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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New post 14 Mar 2018, 12:29
pushpitkc wrote:
prady2231 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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New post 14 Mar 2018, 12:36
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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New post Updated on: 17 Mar 2018, 09:58
1
prady2231 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


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, 20:50.
Last edited by gracie on 17 Mar 2018, 09:58, edited 1 time in total.
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Re: Find the sum of the digit of the least number K  [#permalink]

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New post 16 Mar 2018, 05:39
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prady2231 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


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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New post 17 Mar 2018, 02:31
Can anyone post the link to the topic this question is covering ?
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Re: Find the sum of the digit of the least number K  [#permalink]

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New post 17 Mar 2018, 07:44
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
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Re: Find the sum of the digit of the least number K &nbs [#permalink] 17 Mar 2018, 07:44
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