Question

A gardener has \[1000\] plants. He wants to plant these in such a way that the number of rows and the number of columns remain the same. Find the minimum number of plants he needs more for this.

Answer 1

Hint: As the gardener has to plant \[1000\] plants in a way that the number of rows and columns are equal, find the least perfect square greater than \[1000\]. Subtract \[1000\] from the value of the perfect square number to get the minimum number of plants that we need to arrange all the plants in equal number of rows and columns.

We have to plant \[1000\] plants in such a way that the number of rows and columns are equal. We have to find the minimum number of plants that we need more to do so.
Firstly, we will observe that if plants have to be arranged in a way such that the number of rows and columns are equal, we are arranging plants in a square.
Thus, we need to find a number which is a perfect square next to the number \[1000\].
We observe that \[{{\left( 31 \right)}^{2}}=961\] and \[{{\left( 32 \right)}^{2}}=1024\].
Thus, the least perfect square greater than \[1000\] is \[1024\].
To find the number of plants that we need to plant more, we will subtract \[1000\] from \[1024\].
So, we have \[1024-1000=24\].
Hence, we need to plant at least \[24\] more plants to arrange them in equal numbers of rows and columns. We will arrange those \[1024\] plants in \[32\] rows and columns.

Note: We must keep in mind that we have to find the number of plants that we need more to arrange them in equal number of rows and columns and not the number of plants that we need to remove; otherwise we will get an incorrect answer. A perfect square is a number obtained by multiplying a whole number by itself. The perfect square numbers must end with digits \[1,4,5,6,9\]. Perfect squares never end with digits \[2,3,7,8\].