Question 9

We need to count the total number of tiny squares and then find its prime factorization.

Step 1 — Counting the big squares

First, let us count how many big squares are arranged in the picture. We can count the number of big squares in each row and each column.

Number of big squares in a row = 9

Number of big squares in a column = 9

To find the total number of big squares, we multiply the number of rows by the number of columns.

Total number of big squares = Number of rows $\times$ Number of columns

$= 9 \times 9$

$\boxed{81}$

Step 2 — Counting tiny squares in one big square

Next, let us look closely at one of the big squares. Each big square is made up of many tiny squares. We count the tiny squares along one side of a big square.

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

Since each big square is a grid, the total tiny squares in one big square is $5 \times 5$.

Number of tiny squares in each big square = $5 \times 5$

$= 25$

$\boxed{25}$

Step 3 — Calculating total tiny squares

Now, we can find the total number of tiny squares in the entire picture. We multiply the total number of big squares by the number of tiny squares in each big square.

Total tiny squares = Total number of big squares $\times$ Number of tiny squares in each big square

$= 81 \times 25$

$= 2025$

$\boxed{2025}$

Step 4 — Prime factorization of the total tiny squares

Prime factorization is breaking down a number into its prime factors (numbers only divisible by 1 and themselves). We will find the prime factors of 2025.

We start by dividing 2025 by the smallest prime numbers. Since 2025 ends in 5, it is divisible by 5.

$2025 \div 5 = 405$

$405 \div 5 = 81$

Now we need to factorize 81. We know that $81 = 9 \times 9$. Since 9 is $3 \times 3$, we can write 81 as $3 \times 3 \times 3 \times 3$.

So, the prime factors of 2025 are 5, 5, 3, 3, 3, 3. We can write this using exponents.

Prime factorization of 2025 = $3 \times 3 \times 3 \times 3 \times 5 \times 5$

$= 3^4 \times 5^2$

$\boxed{3^4 \times 5^2}$

Answer

(i) The total number of tiny squares in the picture is 2025. (ii) The prime factorization of the number of tiny squares is $3^4 \times 5^2$.