# TANCET - XAT Math Question : Number Theory

The question given below is a sample practice problem in Number Theory. It is an Arithmetic Topic and the problem provides an understanding of the number of divisors or factors that a number has.

## Question

48 students have to be seated such that each row has the same number of students as the others. If at least 3 students are to be seated per row and at least 2 rows have to be there, how many arrangements are possible?

- 4
- 10
- 8
- 7
- 6

Correct Answer -

**7**. Choice (4)

#### Explanatory Answer

Three conditions have to be satisfied.

- The number of students per row has to be at least 3.
- Number of row has to be at least 2.
- Equal number of students has to be seated in a row.

The following arrangements satisfy all 3 conditions.

**Arrangement 1**: 3 students to a row; 16 rows.

**Arrangement 2**: 4 students to a row; 12 rows.

**Arrangement 3**: 6 students to a row; 8 rows.

**Arrangement 4**: 8 students to a row; 6 rows.

**Arrangement 5**: 12 students to a row; 4 rows.

**Arrangement 6**: 16 students to a row; 3 rows.

**Arrangement 7**: 24 students to a row; 2 rows.

You will observe that the number of students in a row is a factor of 48.

So, an alternative and faster approach is to list down factors of 48 – viz., 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.

And then start from 3 and quickly find out if the number of rows is at least 2.

Both the conditions are satisfied for the following factors : 3, 4, 6, 8, 12, 16 and 24. i.e., 7 arrangements.

Correct answer choice (4)

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