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# problem

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A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row and each column contains the same number of chairs as every other column. If there must be at least two chairs in every row and column and all of the chairs in the room must be included, how many arrays are possible in a classroom containing 36 chairs? Note that 12 rows of 3 chairs is different from 3 rows of 12 chairs.

Jun 26, 2018

#1
+24089
+2

A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row and each column contains the same number of chairs as every other column. If there must be at least two chairs in every row and column and all of the chairs in the room must be included, how many arrays are possible in a classroom containing 36 chairs? Note that 12 rows of 3 chairs is different from 3 rows of 12 chairs.

The divisors of 36 are: 1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 36 (9 divisors)

There must be at least two chairs in every row and column:

The result is:

$$\begin{array}{|l|rcl|} \hline & &\text{array}& \\ \hline 1. & 2 &\times& 18 \\ \hline 2. & 18 &\times& 2 \\ \hline 3. & 3 &\times& 12 \\ \hline 4. & 12 &\times& 3 \\ \hline 5. & 4 &\times& 9 \\ \hline 6. & 9 &\times& 4 \\ \hline 7. & 6 &\times& 6 \\ \hline \end{array}$$

7 arrays are posible in a classroom containing 36 chairs?

Jun 26, 2018

#1
+24089
+2

A rectangular array of chairs is an arrangement of the chairs in rows and columns such that each row contains the same number of chairs as every other row and each column contains the same number of chairs as every other column. If there must be at least two chairs in every row and column and all of the chairs in the room must be included, how many arrays are possible in a classroom containing 36 chairs? Note that 12 rows of 3 chairs is different from 3 rows of 12 chairs.

The divisors of 36 are: 1 | 2 | 3 | 4 | 6 | 9 | 12 | 18 | 36 (9 divisors)

There must be at least two chairs in every row and column:

The result is:

$$\begin{array}{|l|rcl|} \hline & &\text{array}& \\ \hline 1. & 2 &\times& 18 \\ \hline 2. & 18 &\times& 2 \\ \hline 3. & 3 &\times& 12 \\ \hline 4. & 12 &\times& 3 \\ \hline 5. & 4 &\times& 9 \\ \hline 6. & 9 &\times& 4 \\ \hline 7. & 6 &\times& 6 \\ \hline \end{array}$$

7 arrays are posible in a classroom containing 36 chairs?

heureka Jun 26, 2018