Students in a class are arranged to form groups of 4 members each

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?

If the students had been arranged in groups of 4 members each, however, 3 students would be left --> x = 4q + 3 --> x could be 3, 7, 11, 15, 19, 23, 27, 31, ...
If the students had been arranged in groups of 9 members each, however, 4 students would be left --> x = 9p + 4 --> x could be 4, 13, 22, 31, ...

From above general formula would be: x = 36n + 31 --> x could be 31, 67, 103, 139, ... (for more about deriving general formula check: manhattan-remainder-problem-93752.html#p721341)

(1) The number of students is a two-digit number less than 70 --> x could be 31 or 67. Not sufficient,

(2) The number of students is a two-digit number greater than 50 --> x could only be 67. Sufficient.

Answer: B.
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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.

GMAT assassins aren't born, they're made,
Rich

This question frustrated me to the core in veritas cat 7. It took nearly 06m 53s nearly 7 min and I selected the wrong choice out of panic and exhaustion and got it wrong.

I calculated all the values necessary and by the time I'm evaluating statement 2 after 67, I felt exhausted because of calculations and having realized that I spent too much time, I guessed it thereby decreasing my score and question difficulty level.

I don't think we have time in calculating general formula as Bunuel suggested here(Although it is good info to learn) whereas we can go reverse in our approach instead.

Instead of calculating values or N=4x+3 and checking them by other equation 9y+4, we can go by calculating values for 9y+4 since two digit numbers with this equation is possible for only 10 values of y=1 to 9.

the no's will be 13, 22, 31, 40, 49, 58, 67, 76, 85 and 94

checking whether these values satisfy 4x+3 could filter values to 31 and 67.

Disclaimer: Please note that he gave a big hint that it is two digit number in choices combined.
Lets go for choices now.

(1) The number of students is a two-digit number less than 70 --> Both our values are below 70. x could be 31 or 67. Since this DS is value based question (not yes/no type) we need one fixed value and this Not sufficient.

(2) The number of students is a two-digit number greater than 50 --> x could only be 67. Sufficient.

Therefore Answer: B.

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.

Merging topics. Please refer to the discussion above.
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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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x=4n+3: 3,7,11,15…31…67…103…
x=9m+4: 4,13,22,31…67…103…
x=lcm(9,4)=36y
x=36y+31

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

Ans (B)

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.
The number in class can be 31 or 67 - Insufficient

(2) The number of students is a two-digit number greater than 50.
The only 2 digit number greater than 50 which satisfies the condition is 67. B is alone is sufficient.

Ans B
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