What is the prime factorization of 490?
Understand the Problem
The question is asking for the prime factorization of the number 490, which involves expressing 490 as a product of its prime factors.
Answer
The prime factorization of 490 is $2 \times 5 \times 7^2$.
Answer for screen readers
The prime factorization of 490 is $2 \times 5 \times 7^2$.
Steps to Solve
- Start with the number 490
Begin by dividing 490 by the smallest prime number, which is 2, since 490 is even.
$$ 490 \div 2 = 245 $$
- Factor 245 further
Next, take 245 and check the next smallest prime number, which is 3. Since the sum of the digits (2 + 4 + 5 = 11) is not divisible by 3, we move to the next prime, which is 5. 245 ends with 5, so we can divide by 5.
$$ 245 \div 5 = 49 $$
- Factor 49
Now, we have 49. The smallest prime number to check is again 2 (it doesn't work), then 3 (it doesn't work), and then 5 (it doesn't work). The next prime is 7.
$$ 49 \div 7 = 7 $$
- Complete the factorization
We can factor 7 as:
$$ 7 \div 7 = 1 $$
Now, we have all prime factors.
- Write down the prime factorization
Combine all the prime factors we found:
$$ 490 = 2 \times 5 \times 7 \times 7 $$
This can also be written as:
$$ 490 = 2 \times 5 \times 7^2 $$
The prime factorization of 490 is $2 \times 5 \times 7^2$.
More Information
When using prime factorization, we identify the prime numbers that multiply together to give the original number. This method is useful in various areas of mathematics, including simplifying fractions and solving problems related to divisibility.
Tips
- Forgetting to check primes in ascending order can lead to incorrect factors. Always start with the smallest prime.
- Miscalculating the results of divisions. Be careful with each calculation step, especially when dealing with larger numbers.
