Questions: Write 75 as a product of prime factors.

Transcript text: Write 75 as a product of prime factors.

Solution

Solution Steps

To write 75 as a product of prime factors, we need to find the prime numbers that multiply together to give 75. We start by dividing 75 by the smallest prime number (2) and continue with the next smallest primes (3, 5, 7, etc.) until we reach 1.

Step 1: Prime Factorization

To express \( 75 \) as a product of prime factors, we start by dividing \( 75 \) by the smallest prime number, which is \( 2 \). Since \( 75 \) is odd, we move to the next prime number, \( 3 \).

Step 2: Division by Prime Numbers

Dividing \( 75 \) by \( 3 \): \[ 75 \div 3 = 25 \] Next, we factor \( 25 \) by dividing it by the smallest prime number, \( 5 \): \[ 25 \div 5 = 5 \] Finally, we divide \( 5 \) by \( 5 \): \[ 5 \div 5 = 1 \]

Step 3: Collecting Prime Factors

The prime factors we found are \( 3 \) and \( 5 \). Since \( 5 \) appears twice in the factorization, we can express \( 75 \) as: \[ 75 = 3 \times 5 \times 5 = 3 \times 5^2 \]

Final Answer

Thus, the complete prime factorization of \( 75 \) is: \[ \boxed{75 = 3 \times 5^2} \]

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