Question

The dimensions of a window are $156\text{ cm} \times 216\text{ cm}$. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

Hint:

For area division problems, the number of squares can also be found directly by multiplying the simplified division ratios:
$\frac{156}{12} = 13$ squares along the width, and $\frac{216}{12} = 18$ squares along the height.
Total squares $= 13 \times 18 = 234$.

Answer 1

Step 1: Find the H.C.F. of 156 and 216 using the Euclidean algorithm instead of prime factorisation. \[ 216 = 1 \times 156 + 60 \] \[ 156 = 2 \times 60 + 36 \] \[ 60 = 1 \times 36 + 24 \] \[ 36 = 1 \times 24 + 12 \] \[ 24 = 2 \times 12 + 0 \] The last nonzero remainder is $12$, so H.C.F.$(156,216) = 12\text{ cm}$.
Step 2: This H.C.F. is the largest possible square side.
A square grill of side $12\text{ cm}$ will divide both the $156\text{ cm}$ and $216\text{ cm}$ sides exactly, and no larger square can.
Step 3: Count squares along each side. \[ \frac{156}{12} = 13, \qquad \frac{216}{12} = 18 \]
Step 4: Multiply to get the total number of squares. \[ 13 \times 18 = 234 \]
\[ \boxed{\text{Side} = 12\text{ cm}, \ \text{Number of squares} = 234} \]