Question

David has a total of 13 coins. How many different ways will he arrange them in a row with the same number of coins?

• A. 1 x 13 and 13 x 1
• B. 9 x 4 and 4 x 9
• C. 3 x 10 and 10 x 3

Answer 1

The correct option is A 1 x 13 and 13 x 1

David has a total of 13 coins
13 is a prime nmber.
Factors of 13 are 1, 3, 9
1 × 13 = 13
13 × 1 = 9
So, he could have arranged the coins in 1 row and 13 colums or 13 rows and 1 column

Let's C represents coins

1 Row 13 Columns: 1 x 13
C C C C C C C C C C C C C

13 Rows 1 Column: 13 x 1
C
C
C
C
C
C
C
C
C
C
C
C
C

3 Rows 10 Columns: 3 x 10
C C C C C C C C C C
C C
C
Each row does not contain the same number of coins. The first rows have 10 coins, while the last two rows has 2 coins and 1 coin respectively.
10 Rows 3 Columns: 10 x 3
C C C
C C
C
C
C
C
C
C
C
C

Each row does not contains the same number of coins. The first rows have 3 coins, while the last rows have one coin.

Similarly, 9 rows and 4 columns (9 x 4) and 4 rows and 9 columns (4 x 9) are not possible.

Thus, the only possibility is that he could arrange the 13 coins into either 13 rows and 1 column or 1 row and 13 columns and vice versa.