What are the prime factors of 490?
Understand the Problem
The question is asking for the prime factors of the number 490. To solve it, we will find the prime numbers that multiply together to give the number 490.
Answer
The prime factors of 490 are $2$, $5$, and $7^2$.
Answer for screen readers
The prime factors of 490 are $2$, $5$, and $7^2$.
Steps to Solve
- Divide by the smallest prime number
Start with the smallest prime number, which is 2. Check if 490 is divisible by 2.
$$ 490 \div 2 = 245 $$
Since 245 is a whole number, we can say that 2 is a prime factor.
- Continue factoring with the next smallest prime number
Next, we need to factor 245. The smallest prime number after 2 is 3. Check if 245 is divisible by 3.
$$ 245 \div 3 \approx 81.67 $$
Since it is not a whole number, we move to the next prime number, which is 5.
- Check division by 5
Now, check if 245 is divisible by 5.
$$ 245 \div 5 = 49 $$
Since 49 is a whole number, we find that 5 is another prime factor.
- Factor 49
Next, we need to factor 49. The smallest prime that can be tested is 2, but since 49 is odd, we try the next smallest prime, which is 3.
$$ 49 \div 3 \approx 16.33 $$
This isn't a whole number. Next, we check 5.
$$ 49 \div 5 \approx 9.8 $$
Still not a whole number. Finally, we check 7.
$$ 49 \div 7 = 7 $$
Since 7 is a whole number, we find that 7 is also a prime factor, and it appears twice.
- Compile the prime factors
Putting it all together, from the steps taken we have:
$$ 490 = 2 \times 5 \times 7 \times 7 $$
Which can also be written as:
$$ 490 = 2 \times 5 \times 7^2 $$
The prime factors of 490 are $2$, $5$, and $7^2$.
More Information
The number 490 can be factored into its prime components, which illustrates the fundamental theorem of arithmetic: that every integer greater than 1 either is a prime number or can be uniquely represented as a product of prime numbers.
Tips
- Confusing composite numbers with prime numbers: Ensure to recognize that the factors must be prime. For example, stopping at 49 instead of factoring it down to $7 \times 7$ is a common mistake.
- Dividing wrongly: Make sure to do the arithmetic correctly when dividing by prime numbers.
