Students in a class are arranged to form groups of 4 members each. After forming the groups, 3 students are left. If the students had been arranged in groups of 9 members each, however, 4 students would be left. What is the total number of students in the class?

(1) The number of students is a two-digit number less than 70.
(2) The number of students is a two-digit number greater than 50.

I got the answer to this question, but is there any easy way to find the answer other than by listing all the numbers?
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Hi TARGET730,

For this DS question, you really do have to list out a certain number of the possibilities (until you have enough information that you know that you can stop working).

The reason why is because the algebra equations that you could form from the information in the prompt do not provide enough information for you to solve.

Z = Total number of students

If students are formed into groups of 4 members each, then 3 students would be left...

4X + 3 = Z

If students are formed into groups of 9 members each, then 4 students would be left..

9Y + 4 = Z

Here, we have 3 variables, but only 2 equations, so while we can establish a relationship between Y and X (by combining the 2 equations), we can't figure out the exact value of either one until we "map out" the possibilities.

We'd end up with

4X + 3 = 9Y + 4
4X = 9Y + 1

Knowing that X and Y have to be integers limits the possibilities, but we'd still have to list them out.

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Rich
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Students in a class are arranged to form groups of 4 members each. After forming the groups, 3 students are left. If the students had been arranged in groups of 9 members each, however, 4 students would be left. What is the total number of students in the class?

(1) The number of students is a two-digit number less than 70.

(2) The number of students is a two-digit number greater than 50.
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We are given that when we divide the total students by 4, we have a remainder of 3, so the total could be values such as:

7, 11, 15, 19, 23, 27, 31, 35, 39, ...

We are given that when we divide the total students by 9, we have a remainder of 4, so the total could be values such as:

13, 22, 31, 40, 49, 58, ...

The first value common to our lists is 31; we can keep adding the LCM of 4 and 9, which is 36, to 31 to generate succeeding values. Thus, the total number of students could be:

31, 67, 103, ...


We need to determine the exact total number of students.

Statement One Alone:

The number of students is a two-digit number less than 70.

Since the total number of students could be 31 or 67, statement one alone is not sufficient to answer the question.

Statement Two Alone:

The number of students is a two-digit number greater than 50.

Of the numbers 31, 67, 103, ... there is only one number that is both a 2-digit number and greater than 50. Thus, statement two is sufficient.

Answer: B
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