Question

A farmer decides to build an orchard with 32 orange trees, 24 mango trees and 40 pear trees. If each row has a similar type of fruit tree and equal number of fruit trees, the minimum number of possible rows is .

Answer 1

We know,
$\text{Total number of trees}=\text{Number of rows}\times \text{Number of columns}$

The total number of trees is fixed.
$\therefore$ Minimum number of rows of tree results in maximum number of tree in each row.

Also, each row contains the same type of fruit tree.

$\therefore$ We need to find the greatest common factor of $32,24$ and $40.$

Using prime factorization,
$32=2\times 2\times 2\times 2\times 2$
$40=2^3\times 5$
$24=2\times 2\times 2\times 3$

$\Rightarrow$ GCD $=8$

$\therefore$ Required number of fruit trees in each row is $8.$

Rows of orange trees are 4.
Rows of mango trees are 5.
Rows of pear trees are 3.

Hence, total number of fruit trees are $4+5+3=12$