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n and p are integers greater than 1 5n is the square of a number 75np

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Bunuel
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n and p are integers greater than 1 5n is the square of a number 75np [#permalink] New post   22 May 2016, 13:55
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n and p are integers greater than 1;
5n is the square of a number;
75np is the cube of a number.
The smallest value for n + p is

A. 14
B. 18
C. 20
D. 30
E. 50
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DensetsuNo
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n and p are integers greater than 1 5n is the square of a number 75np [#permalink] New post   22 May 2016, 14:19
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\(n > 1\)
\(p > 1\)
\(5*n = x^2\)
\(75np = y^3\)
\(n+p = ?\)

Well, a square number is usually the result of himself times himself so in the case of n the least value possible (being 5 a prime) is 5*5
\(n=5\)
as for p
\(75*5*p= y^3\)
we know \(375\) is equal = \(5^3*3^1\) therefore to obtain a cube we need \(3^2\) or 9 -> \(p=9\)

\(p+n=14\)

A)
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Re: n and p are integers greater than 1 5n is the square of a number 75np [#permalink] New post   18 Jul 2017, 20:15
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n and p are integers greater than 1; 5n is the square of a number; 75np is the cube of a number. The smallest value for n + p is

Given: 5n is the square of a number
This means that n=5\(a^2\), where a>=1

Given: 75np is the cube of a number
This means that 3*25*5\(a^2\)p is the cube of a number. This implies that p=\(3^2\)a

Now n+p = 5\(a^2\) + 9a = a [5a+9]

The smallest value of n+p, will be when a=1. Therefore, n+p = 5+9 = 14

Answer is A
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Re: n and p are integers greater than 1 5n is the square of a number 75np [#permalink] New post   21 Mar 2020, 15:43
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5n = x^2
n = 5 because in order for x^2 to be an integer square, , the root must also be an integer

75np = x^3
75 = 3*5*5
75n = 3*5*5*5

in order for x^3 to be cubed, we need an even distribution of factors.
75n has three 5's and one 3, so p needs to add an additional two 3's

75np = 3*3*3*5*5*5

therefore:
n=5
p=3*3=9

n+p = 14
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Re: n and p are integers greater than 1 5n is the square of a number 75np [#permalink] New post   17 Mar 2020, 09:58
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Bunuel wrote:
n and p are integers greater than 1;
5n is the square of a number;
75np is the cube of a number.
The smallest value for n + p is

A. 14
B. 18
C. 20
D. 30
E. 50


The smallest value for n is 5.

Recall that a perfect cube has prime factors that must each be raised to a multiple of 3. We will use this fact to solve for p.

So 75np = 375p, and since 375p = 5^3 x 3p is a cube, we see that the smallest value for p is 9 (notice that 375p = 5^3 x 3^3). Therefore, the smallest value for n + p is 5 + 9 = 14.

Answer: A
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Re: n and p are integers greater than 1 5n is the square of a number 75np [#permalink]
17 Mar 2020, 09:58
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