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 division method instead of prime factorisation.
Divide 216 by 156: $216 = 1\times156+60$. Divide 156 by 60: $156=2\times60+36$. Divide 60 by 36: $60=1\times36+24$. Divide 36 by 24: $36=1\times24+12$. Divide 24 by 12: $24=2\times12+0$.
Step 2: Read off the H.C.F.
The last nonzero remainder is $12$, so the biggest square side length is $12\text{ cm}$.
Step 3: Count squares along each side of the window separately.
Along the $156\text{ cm}$ side: $\frac{156}{12}=13$ squares. Along the $216\text{ cm}$ side: $\frac{216}{12}=18$ squares.
Step 4: Multiply to get the total count and conclude.
\[ \text{Total squares} = 13 \times 18 = 234 \]
\[ \boxed{\text{Side} = 12\text{ cm},\ \text{Number of squares} = 234} \]