Question

Number of different squares of any size (side of square be natural number) which can be made from a rectangle size \[15 \times 8\] is

Answer 1

Hint: Here in this question we have to form the squares of different dimensions from the rectangle, whose dimension is \[15 \times 8\]. By using the formula \[S = \dfrac{1}{6}(N)(N + 1) \times (3M - N + 1)\], where M represents the value of length of a rectangle i.e., 15 and N represents the value of breadth of a rectangle. On substituting these values in the formula and simplifying we obtain the required solution.

Complete step-by-step answer:
A rectangle and a square are a plane figure. A rectangle is a four sided-polygon. The two sides at each corner or vertex meet at right angles. The opposite sides of the rectangle are equal in length. A rectangle is also a two-dimensional figure.
A square is a two-dimensional plane figure with four equal sides and all the four angles are equal to \[{90^ \circ }\]. The properties of a rectangle are similar to a square, but the difference between the two is, a rectangle has only its opposite sides equal. Therefore, a rectangle is called a square only if all its four sides are of equal length.
Here we have a rectangle whose length is 15 units and the breadth is 8 units. In this rectangle we have to form a square and it can be different dimensions.
By using the formula \[S = \dfrac{1}{6}(N)(N + 1) \times (3M - N + 1)\], where \[M \geqslant N\]. Here, the value of \[M = 15\] and the value of \[N = 8\]. S is the number of squares.
On substituting these values in the above formula.
\[ \Rightarrow S = \dfrac{1}{6}(8)(8 + 1) \times (3 \times 15 - 8 + 1)\]
On simplifying we have
\[ \Rightarrow S = \dfrac{1}{6}(8)(9) \times (45 - 8 + 1)\]
\[ \Rightarrow S = \dfrac{1}{6} \times 72 \times 38\]
On dividing the number 72 by 6
\[ \Rightarrow S = 12 \times 38\]
On multiplying the number 12 and 38
\[ \Rightarrow S = 456\]
Therefore the number of different squares of any size which can be made from a rectangle size \[15 \times 8\] is 456.
So, the correct answer is “456”.

Note: The properties of a square and rectangle are almost similar to each other. When we are drawing a square from the rectangle, then the maximum length of a square will be equal to the breadth of the rectangle and the least length of square will be one unit. Likewise we can also make a rectangle from a square.