If 72 cupcakes must be divided equally among the students in a certain class, how many students are in the class?

(1) If the number of students is reduced by one-third, each student will receive 3 more cupcakes.

(2) If the number of cupcakes is doubled, each student will receive 12 cupcakes.
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Sky is the limit

Straight and direct question: "D"

D is the answer. Two approaches to solve this:

Approach 1: Simple Algebra
Let 'n' be the number of students in the class. Let 'c' be the number of cupcakes each student receives. We are asked to find the value of 'n'.
We know that n*c = 72.

Statement 1: Number of students is reduced by 1/3. Thus number of students becomes '2n/3'.
Also, each student now receives 3 more cupcakes. Thus the number of cupcakes that each student receives = (c + 3).
Again (2n/3)(c + 3) = 72. Using this equation with the one obtained previously, we have
nc = (2n/3)(c + 3). Solving this we can get a value of 'c', which can further give us a value of 'n'. Thus SUFFICIENT.

Statement 2: Number of cupcakes is doubled. Thus number of cupcakes now = 72 * 2 = 144
Each student now receives 12 cupcakes. Thus c = 12. Substitute in the equation nc = 144 to find the value of 'n'. Thus SUFFICIENT.

Thus each statement is SUFFICIENT.

Approach 2: Factors
72 cupcakes must be divided equally among the students. Thus, if the number of students in the class is 'n', and the number of cupcakes that each student receives is 'c', then we know that 'n' and 'c' both are factors of 72.
We can find out factor pairs of 72 as below:
1 * 72
2 * 36
3 * 24
4 * 18
6 * 12
8 * 9

So 'n' and 'c' both will have values from among the factors listed above.
Statement 1:
Each student now receives 3 more cupcakes. Thus, we can infer that even (c + 3) is a factor of 72. The numbers that satisfy the condition that both c and (c + 3) are factors of 72 are: c = 1, 3, 6, 9.
By having a look at the corresponding pair of factors of the above numbers, we can find that the values of 'n' can be 72, 24, 12 and 8 respectively. That is, if c = 1, n = 72; if c = 3, n = 24 and so on.

Number of students is reduced by 1/3. Thus the new number of students (2n/3) is also a factors of 72. Now from our analysis above, we know that 8 cannot be the value of 'n' because if n = 8, then (2n/3) is not an integer. This is not possible as the number of students has to be an integer.
If we take n = 12, then we find that 2n/3 = 8. Thus c = 9. This agrees with our analysis that initially n = 12; new n = 8; previously c = 6; new c = 9. Thus SUFFICIENT.

Statement 2:
Number of cupcakes now = 144. Each student receives 2c. Thus 2c = 144. Thus, c = 12. Put this in the original equation to find a value of 'n'. Thus SUFFICIENT.

Thus both statements are SUFFICIENT.

Though the analysis for the second approach is a bit longer, it takes less than 30 seconds to solve using that method. The longer explanation need not imply longer time to think and analyze.