Rational Numbers | FIO

Question 9

How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.

Solution

Understand the Question

The given image consists of a large grid arranged into $9$ rows and $9$ columns of big squares.
Each big square is further divided into a smaller grid of $5 \times 5$ tiny squares.
To find the total number of tiny squares, multiply the total number of big squares by the number of tiny squares in each big square.
The prime factorisation is obtained by expressing the total count as a product of prime powers.

Step 1 · Count Total Number of Big Squares

Number of big squares in a row $= 9$ Number of big squares in a column $= 9$ Diagram 1

\begin{aligned} \text{Total number of big squares} &= \text{Number of rows} \times \text{Number of columns} \\ &= 9 \times 9 \\ &= 81 \end{aligned}

Step 2 · Count Tiny Squares in Each Big Square

Number of tiny squares along one side of a big square $= 5$

\begin{aligned} \text{Number of tiny squares in each big square} &= 5 \times 5 \\ &= 25 \end{aligned}

Step 3 · Calculate Total Number of Tiny Squares

\begin{aligned} \text{Total tiny squares} &= \text{Total number of big squares} \times \text{Number of tiny squares in each big square} \\ &= 81 \times 25 \\ &= 2025 \end{aligned}

Step 4 · Find Prime Factorisation of Total Tiny Squares

Dividing $2025$ by prime factors:

\begin{aligned} 2025 \div 5 &= 405 \\ 405 \div 5 &= 81 \end{aligned}

Since $81 = 9 \times 9 = (3 \times 3) \times (3 \times 3)$ :

\begin{aligned} \text{Prime factorisation of } 2025 &= 3 \times 3 \times 3 \times 3 \times 5 \times 5 \\ &= 3^4 \times 5^2 \end{aligned}

Answer

Total tiny squares $= 2025$ ; Prime factorisation $= 3^4 \times 5^2$

Common Mistakes

Manual Counting: Attempting to count all tiny squares individually instead of multiplying grid dimensions ( $81 \times 25$ ).
Incomplete Prime Factorisation: Leaving factors with composite bases like $9^2 \times 5^2$ or $81 \times 25$ instead of reducing $81$ to prime base $3^4$ .
Direct Factorisation Shortcut: $2025 = 81 \times 25 = 3^4 \times 5^2$ can be factored directly from the sub-counts without needing long division.

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