Combining the prime factors, we write 3825 = 3² × 5² × 17. Thus, the exponent of the prime factor 3 is 2.

What is the exponent of $3$ in the prime factorisation of $3825$ ?

Answer: 2

Step-by-step solution

Step 1: Check divisibility by 3 and divide

To find the prime factorisation of $3825$ , we first test divisibility by $3$ . The sum of the digits is $3 + 8 + 2 + 5 = 18$ , which is divisible by $3$ . Dividing $3825$ by $3$ gives $1275$ .

Step 2: Divide the quotient by 3 again

Now we test whether $1275$ is also divisible by $3$ . The sum of the digits is $1 + 2 + 7 + 5 = 15$ , which is divisible by $3$ . Dividing $1275$ by $3$ yields $425$ .

Step 3: Check if 3 divides 425

The sum of the digits of $425$ is $4 + 2 + 5 = 11$ , which is not divisible by $3$ . Therefore, $3$ is no longer a factor of $425$ . We can factorise $425$ as $5^2 \times 17$ .

Step 4: Express in prime power notation and identify exponent

Combining the prime factors, we write $3825 = 3^2 \times 5^2 \times 17$ . Thus, the exponent of the prime factor $3$ is $2$ .

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