Question

6 A rectangular arrangement of pens has rows and columns. Rohan takes away 3 rows of [3] pens and then Sarah takes away 2 columns of pens from the remaining pens. The remaining pens are rearranged in p rows and q columns where p is a prime number.

If Rohan takes 24 pens and Sarah takes 18 pens, find all possible value(s) of p. Show your work.

Answer 1

The possible values of $p$ are 2 and 3.

Answer 2

Let the original arrangement be $M$ rows and $N$ columns.

1. Rohan’s removal: He removes 3 entire rows.
   Since he takes 24 pens,

   $$3 \times N = 24 \implies N = 8.$$

2. Sarah’s removal: After Rohan’s removal, there are $(M-3)$ rows left. She removes 2 entire columns, taking

   $$2 \times (M-3) = 18 \implies M-3 = 9 \implies M = 12.$$

3. Total pens and remaining pens:
   Original number of pens: $12 \times 8 = 96$.
   Pens removed: $24 + 18 = 42$.
   Remaining pens: $96 - 42 = 54$.

4. Rearrangement: The 54 pens are rearranged in $p$ rows and $q$ columns where $p \times q = 54$ and $p$ is a prime number.
   The prime factors of 54 (which factors as $2 \times 3^3$) that can serve as $p$ are:

   • For $p=2$: $q = \frac{54}{2} = 27$.
   • For $p=3$: $q = \frac{54}{3} = 18$.

Answer:
The possible values of $p$ are 2 and 3.