What are the prime factors of 735?
Understand the Problem
The question is asking for the prime factorization of the number 735, meaning we need to find the prime numbers that multiply together to get 735.
Answer
The prime factorization of 735 is $3 \times 5 \times 7^2$.
Answer for screen readers
The prime factorization of 735 is $3 \times 5 \times 7^2$.
Steps to Solve
- Start with the given number
We begin with the number 735. We want to break it down into its prime factors.
- Check for divisibility by the smallest prime (2)
Since 735 is an odd number, it is not divisible by 2. Move to the next smallest prime, which is 3.
- Divide by 3
To check if 735 is divisible by 3, we add the digits: $7 + 3 + 5 = 15$. Since 15 is divisible by 3, we can divide:
$$ 735 \div 3 = 245 $$
- Factor 245
Now, we need to factor 245. Check if it is divisible by 3. Adding the digits: $2 + 4 + 5 = 11$ (not divisible by 3). Next, check 5. The last digit is 5, so it is divisible:
$$ 245 \div 5 = 49 $$
- Factor 49
Next, factor 49. Check for 2 and 3, both not valid. Check 7:
$$ 49 \div 7 = 7 $$
Now we have 7 left.
- Complete the factorization
We have completed the factorization of 735:
$$ 735 = 3 \times 5 \times 7 \times 7 $$
Which can also be expressed as:
$$ 735 = 3 \times 5 \times 7^2 $$
The prime factorization of 735 is $3 \times 5 \times 7^2$.
More Information
Prime factorization is essential for simplifying fractions, finding least common multiples, and solving equations. The number 735 can be practically broken down into smaller prime components that can help in various mathematical applications.
Tips
- Mistaking composite factors for prime factors: Always check if a number can be divided further.
- Not checking for divisibility by successive primes correctly, which may lead to incomplete factorization.
