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A school has \(a\) students and \(b\) teachers. if \(a<15

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A school has \(a\) students and \(b\) teachers. if \(a<15

by AAPL » Fri May 24, 2019 6:40 pm

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A school has \(a\) students and \(b\) teachers. If \(a < 150\), \(b < 25\), and classes have a maximum of 15 students, can the \(a\) students
be distributed among the \(b\) teachers so that each class has the same number of students? (Assume that any student can be
taught by any teacher)

1) It is possible to divide the students evenly into groups of 2, 3, 5, 6, 9, 10, or 15.
2) The greatest common factor of \(a\) and \(b\) is 10.

OA E
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Source: — Data Sufficiency |

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by Ian Stewart » Sat May 25, 2019 9:30 am

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From Statement 1, the number of students, a, is divisible by every number in that list, so it is divisible by the LCM of those numbers. If we just identify the prime divisors in that list, and their relevant exponents, we have a 2, a 5, and 3^2, so the LCM is 90. So the number of students is a multiple of 90, and since that number is less than 150, Statement 1 tells us we have exactly 90 students.

Using that information with Statement 2, we know 10 is divisor of the number of teachers. Since we have less than 25 teachers, we have either 10 or 20 teachers.

So using both statements, we either have 90 students and 10 teachers, and the students can be divided up into equal groups of nine, or we have 90 students and 20 teachers, and we can't divide them equally among the teachers. So the answer is E.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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