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

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Math Expert
Joined: 02 Sep 2009
Posts: 64144
n and p are integers greater than 1 5n is the square of a number 75np  [#permalink]

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22 May 2016, 12:55
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45% (medium)

Question Stats:

70% (01:53) correct 30% (02:26) wrong based on 170 sessions

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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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Posts: 99
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n and p are integers greater than 1 5n is the square of a number 75np  [#permalink]

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22 May 2016, 13:19
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3
$$n > 1$$
$$p > 1$$
$$5*n = x^2$$
$$75np = y^3$$
$$n+p = ?$$

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]

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18 Jul 2017, 19:15
1
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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

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Re: n and p are integers greater than 1 5n is the square of a number 75np  [#permalink]

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17 Mar 2020, 08:58
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.

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Re: n and p are integers greater than 1 5n is the square of a number 75np  [#permalink]

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21 Mar 2020, 14:43
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
Re: n and p are integers greater than 1 5n is the square of a number 75np   [#permalink] 21 Mar 2020, 14:43