NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 1(iii)

NCERT Class X Chapter 1: Real Numbers Exercise 1.1 Question 1(iii)

Question:

Express 3825 as a product of its prime factors.

Given:

The number 3825.

To Find:

Prime factorization of 3825.

Formula:

Successively divide the given number by prime numbers to express it as a product of its prime factors.

Solution:

Step 1: Check divisibility by 2.
3825 is odd, so it is not divisible by 2.

Step 2: Check divisibility by 3.
Sum of digits: 3 + 8 + 2 + 5 = 18, which is divisible by 3.
Divide by 3:

$$ 3825 \div 3 = 1275 $$

Step 3: Check divisibility of 1275 by 3.
Sum of digits: 1 + 2 + 7 + 5 = 15, which is divisible by 3.
Divide by 3:

$$ 1275 \div 3 = 425 $$

Step 4: Check divisibility of 425 by 5.
425 ends with 5, so it is divisible by 5.
Divide by 5:

$$ 425 \div 5 = 85 $$

Step 5: Check divisibility of 85 by 5.
85 ends with 5, so it is divisible by 5.
Divide by 5:

$$ 85 \div 5 = 17 $$

Step 6: Check if 17 is prime.
17 has only two factors: 1 and 17.
So, 17 is a prime number.

Step 7: Write all prime factors together.
$$ 3825 = 3 \times 3 \times 5 \times 5 \times 17 $$

Step 8: Express repeated factors using exponents.
$$ 3825 = 3^2 \times 5^2 \times 17 $$

Result:

3825 as a product of its prime factors is:
$$3825 = 3^2 \times 5^2 \times 17$$

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