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# If k is a positive integer such that 5,880k is a perfect square, what

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Math Expert
Joined: 02 Sep 2009
Posts: 47157
If k is a positive integer such that 5,880k is a perfect square, what  [#permalink]

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20 Mar 2018, 23:19
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45% (medium)

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68% (00:38) correct 32% (01:34) wrong based on 50 sessions

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If k is a positive integer such that 5,880k is a perfect square, what is the least possible value of k?

(A) 2
(B) 6
(C) 15
(D) 30
(E) 140

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Re: If k is a positive integer such that 5,880k is a perfect square, what  [#permalink]

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21 Mar 2018, 00:49
1
IMO D

5880 = 2^3 * 7^2 * 3 * 5

Therefore, to make it a perfect square, the minimum value of k should be = 2 * 3 * 5 = 30

Thus, D.

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Re: If k is a positive integer such that 5,880k is a perfect square, what  [#permalink]

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21 Mar 2018, 02:32
Bunuel wrote:
If k is a positive integer such that 5,880k is a perfect square, what is the least possible value of k?

(A) 2
(B) 6
(C) 15
(D) 30
(E) 140

Bunuel there seems something wrong with the question. Is the integer at hand 5,880*k or 5,880k which is a concatenation of 5,880 & k. Is this an ambiguity or am I making a mistake in reading it properly.

Best,
Math Expert
Joined: 02 Sep 2009
Posts: 47157
Re: If k is a positive integer such that 5,880k is a perfect square, what  [#permalink]

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21 Mar 2018, 02:46
Bunuel wrote:
If k is a positive integer such that 5,880k is a perfect square, what is the least possible value of k?

(A) 2
(B) 6
(C) 15
(D) 30
(E) 140

Bunuel there seems something wrong with the question. Is the integer at hand 5,880*k or 5,880k which is a concatenation of 5,880 & k. Is this an ambiguity or am I making a mistake in reading it properly.

Best,

5,880k is 5,880*k, so 5,880 multiplied by k (also recall that multiplication sign is often omitted). It cannot be a five-digit integer 5880k because if it were it would have been explicitly mentioned.
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Re: If k is a positive integer such that 5,880k is a perfect square, what  [#permalink]

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22 Mar 2018, 15:47
Bunuel wrote:
If k is a positive integer such that 5,880k is a perfect square, what is the least possible value of k?

(A) 2
(B) 6
(C) 15
(D) 30
(E) 140

We are given that 5,880k is a perfect square. We must remember that all perfect squares break down to unique prime factors, each of which has an exponent that is a positive multiple of 2. So, let’s break down 5,880 into its prime factors to determine the minimum value of k.

5,880 = 588 x 10 = 12 x 49 x 10 = 2^3 x 3^1 x 5^1 x 7^2

So, in order for 5,880k to be a perfect square, k must contain the factors 2 x 3 x 5 = 30 (so that 5,880k = 2^4 x 3^2 x 5^2 x 7^2), so 30 is the smallest possible value of k.

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If k is a positive integer such that 5,880k is a perfect square, what  [#permalink]

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25 Mar 2018, 13:31
1

Solution

• $$5880* k$$ can be written in prime factorization form as $$2^3 * 3 *5 * 7^2 * k$$.

• Since $$5880 * k$$ is a perfect square, hence $$5880 * k=n^2$$, where $$n$$ is any integer.
o $$n^2 = 2^3 * 3 * 5 * 7^2 * k$$

o After taking square root on both the sides, we can write:
o $$n= 2*7* \sqrt{(2*3*5*k)}$$
oFor $$n$$ to be an integer,$$\sqrt{(2 * 3 * 5 * k)}$$ must be a perfect square.
oHence,$$k = 2 * 3 * 5 *$$ (square of any positive integer)
.

• The least possible value of the square of a positive integer is 1.
o Hence, k=2 * 3 * 5 = 30
.

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Re: If k is a positive integer such that 5,880k is a perfect square, what  [#permalink]

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26 Mar 2018, 00:22
Bunuel wrote:
If k is a positive integer such that 5,880k is a perfect square, what is the least possible value of k?

(A) 2
(B) 6
(C) 15
(D) 30
(E) 140

The rule says, " A perfect square always has even number of powers of prime factors".

Prime factorisation of 5880 = 2^3*3^1*5^1*7^2

Therefore, to make 5880 a perfect square we need to multiply it by a set of 2*3*5 = 30

Hence (D)
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Re: If k is a positive integer such that 5,880k is a perfect square, what &nbs [#permalink] 26 Mar 2018, 00:22
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