Chapter 4.1, Problem 56E

To determine

To find out the number of possible displays that can be made with the same number fireflies in each row.

Answer to Problem 56E

3 rows of 23 fireflies each or 23 rows of 3 fireflies each

Explanation of Solution

Given Information: The total number of different species of fireflies as 69.

Formula used: By calculating the prime factors of the given number, the number of rows and the equal number of fireflies in each row may be evaluated.

Calculation:

At first, find the prime factor of the number 69 to identify the number of rows and equal number of fireflies for each row, that make up the number 69.

To get the Prime Factors of 69, you divide 69 by the smallest prime number possible. Then you take the result from that and divide that by the smallest prime number. Repeat this process until you end up with 1.

Here is the math to illustrate:

  $\begin{array}{l}69÷3=23\\ 23÷23=1\end{array}$

Again, all the prime numbers you used to divide above are the Prime Factors of 69. Thus, the Prime Factors of 69 are:

3, 23.

There are two conclusions to this answer as:

Since the number 69 can be made commutatively by these two numbers: 3 and 23

i.e. $3\times 23|23\times 3$

Hence, there are two possibilities as

EITHER there are 3 rows of 23 fireflies each

ORthere are 23 rows of 3 fireflies each