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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