A nursery worker has 72 small trees to be displayed in lines in a rectangular shape. Each horizontal and vertical line must have more than 5 trees. How many different arrangements are possible?

Question

A nursery worker has 72 small trees to be displayed in lines in a rectangular shape. Each horizontal and vertical line must have more than 5 trees. How many different arrangements are possible?

1 Answer

Impact of this question

1071 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-26T02:30:19

2 arrangements
$6 \cdot 12$ $6*12$
$8 \cdot 9$ $8*9$

Explanation:

The important part of this question is to recognize factor pairs of 72 that are both larger than 5.

By prime factorization, we have
$72 = 2^{3} \cdot 3^{2}$ $72=2^3*3^2$ ,
Which gives a total of $(3 + 1) (2 + 1) = 12$ $(3+1)(2+1)=12$ factors, and hence 6 pairs of factors.

Note that 1, 2, 3, 4 (which is $2 \cdot 2$ $2*2$ ) are smaller than 5, so we have to deduct 4 pairs, giving the answer of 2 pairs.

Method 2: Listing

$72$ $72$
$= 1 \cdot 72$ $=1*72$
$= 2 \cdot 36$ $=2*36$
$= 3 \cdot 24$ $=3*24$
$= 4 \cdot 18$ $=4*18$
$= 6 \cdot 12$ $=6*12$
$= 8 \cdot 9$ $=8*9$

As we can see, there are only $6 \cdot 12$ $6*12$ and $8 \cdot 9$ $8*9$ that satisfy the requirement.

Science

Math

Humanities

... and beyond

A nursery worker has 72 small trees to be displayed in lines in a rectangular shape. Each horizontal and vertical line must have more than 5 trees. How many different arrangements are possible?

1 Answer

Explanation:

Related questions

Impact of this question