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An artist creates mosaics using stone tiles provided by his clients. A

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An artist creates mosaics using stone tiles provided by his clients. A  [#permalink]

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New post 15 Aug 2018, 08:38
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A
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D
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An artist creates mosaics using stone tiles provided by his clients. A mathematician challenges him to make a series of large mosaics and a series of small mosaics following a few mathematical restrictions. Each large mosaic must have 85 tiles and each small mosaic must have 75 tiles. The total number of tiles in each series must be identical. What is the total number of mosaics the artist will create?

(1) The total number of mosaics in the series is less than 50.

(2) The number of small mosaics in the series is a prime number.
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Re: An artist creates mosaics using stone tiles provided by his clients. A  [#permalink]

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New post 17 Aug 2018, 11:15
hi,

Can I have the OA please ?

I think it is A.

Since the LCM of 85 &75 is 1275. And only 15 and 17 satisfy condition A.

Please let me know.

Thanks
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Re: An artist creates mosaics using stone tiles provided by his clients. A  [#permalink]

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New post 17 Aug 2018, 12:50
The answer is D I feel.

Once you take the LCM of 85 and 75 which is 1275, it can be obtained that 15 large tiles and 17 small tiles can be made. Only then will both series have the same number of tiles.

A) the total is less than 50. Only possible values are 17 and 15 for a total of 32. - sufficient

B) the number of small tiles is a prime number. Since 17 is a prime number and all multiples will become non-prime, the only possible combination is 15 large tiles and 17 small tiles. - sufficient.

Therefore D is the correct answer.

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Re: An artist creates mosaics using stone tiles provided by his clients. A  [#permalink]

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New post 18 Aug 2018, 04:39
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mikehaueisen wrote:
An artist creates mosaics using stone tiles provided by his clients. A mathematician challenges him to make a series of large mosaics and a series of small mosaics following a few mathematical restrictions. Each large mosaic must have 85 tiles and each small mosaic must have 75 tiles. The total number of tiles in each series must be identical. What is the total number of mosaics the artist will create?

(1) The total number of mosaics in the series is less than 50.

(2) The number of small mosaics in the series is a prime number.


Official Solution (Credit: Manhattan Prep)



Begin by listing the restrictions given in this Linear Equations problem:

Lg M = 85 tiles / mosaic

Sm M = 75 tiles / mosaic

Total # tiles for Lg M series = Total # tiles for Sm M series


Consider how to translate these restrictions into algebra. Create an expression for the total number of tiles used in each series. Let L be the number of large mosaics and S be the number of small mosaics.

Tiles Used in Large Mosaics = 85L

Tiles Used in Small Mosaics = 75S

The number of tiles used in each series must be the same. Also note that L and S must both be positive integers.

85L = 75S

Q: L + S = ?



(1) SUFFICIENT: The number of mosaics in both series must be a positive integer. To test Statement (1), you need to figure out how many values for the total number of mosaics less than 50 will produce an integer value for both S and L. To efficiently determine which values will work use Divisibility & Primes logic. For 85L to equal 75S, the same prime numbers must be present on both sides of the equation. That is the only way that the two products will be equal.

Break each side of the equation down into prime numbers to determine which primes must be present in the variables:

85L = 75S Divide both sides by 5

17L = 15S Break into primes

17 × L = 3 × 5 × S

For the two sides to equal each other, L must be a multiple of 3 and 5 while S must be a multiple of 17. The lowest values that fit these requirements are S = 17 and L = 15, for a total of 32 mosaics. The next smallest multiples of 17 and 15 are 34 and 30, which would imply there are more than 50 mosaics. Therefore, there must be 32 mosaics. Statement (1) is SUFFICIENT. Eliminate choices (B), (C), and (E).


(2) SUFFICIENT: If S is prime and it must be a multiple of 17 for the reasons given in Statement (1), it must be 17. If S is 17, L must be 15. Statement (2) is SUFFICIENT. Eliminate choice (A).


The correct answer is (D).
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Re: An artist creates mosaics using stone tiles provided by his clients. A   [#permalink] 18 Aug 2018, 04:39
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