BTGmoderatorLU wrote:Students in a class are arranged to form groups of 4 members each. After forming the groups, 3 students are left. What is the total number of students in the class?

(1) The number of students is a two-digit number less than 65.
(2) The number of students is a two digit number greater than 30.

Let T = the total number of students.
Since Statement 1 constrains T to a value less than 65, determine whether it's possible that T<65 or that T≥65.

Students in a class are arranged to form groups of 4 members each. After forming the groups, 3 students are left.
Thus, T is equal to 3 more than a multiple of 4:
T = 4a + 3, where a is a nonnegative integer.

If the students had been arranged in groups of 9 members each, however, 4 students would be left.
Thus, T is equal to 4 more than a multiple of 9:
T = 9b + 4, where b is a nonnegative integer.

Since the value of T must be the same in each case, the expressions in blue must be equal:
4a + 3 = 9b + 4
4a = 9b + 1
a = (9b+1)/4.

Since a must be a nonnegative integer, 9b+1 must be a multiple of 4. yielding the following cases:
b=3, with the result that a = (9*3 + 1)/4 = 7 and T = 4a + 3 = (4*7) + 3 = 31.
b=7, with the result that a = (9*7 + 1)/4 = 16 and T = 4a + 3 = (4*16) + 3 = 67.
Since T<65 in the first case but T>65 in the second case, move onto the statements.

Statement 1:
As shown above, the only option for T such that T<65 is T=31.
SUFFICIENT.

Statement 2:
Here, it's possible that T=31 or that T=67.
Since T can be different values, INSUFFICIENT.

The correct answer is A.