Find the prime factorization of 2205.
Understand the Problem
The question is asking to determine the prime factors of the number 2205. We'll break down the number into its prime components through division by prime numbers.
Answer
$3^2 \times 5 \times 7^2$
Answer for screen readers
The prime factorization of 2205 is $3^2 \times 5 \times 7^2$.
Steps to Solve
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Start with the Number We need to factor the number 2205.
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Check for the Smallest Prime Number First, we check if 2205 is divisible by the smallest prime number, which is 2. Since 2205 is odd, it is not divisible by 2. So we move to the next prime number, which is 3.
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Divisibility by 3 To see if it is divisible by 3, we add the digits: $2 + 2 + 0 + 5 = 9$. Since 9 is divisible by 3, we divide: $$ 2205 \div 3 = 735 $$
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Continue Factoring with 3 Now we factor 735 further by 3: Checking the sum of the digits again: $7 + 3 + 5 = 15$. Since 15 is divisible by 3, we divide: $$ 735 \div 3 = 245 $$
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Factor 245 Next, we check if 245 is divisible by the next primes starting from 3. It isn't, let's try 5 (since its last digit is 5): $$ 245 \div 5 = 49 $$
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Factor 49 Finally, we factor 49, which is $7 \times 7$ or $7^2$.
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Combine Prime Factors Now we combine all the prime factors we found: $3, 3, 5,$ and $7, 7$.
Thus, the complete prime factorization of 2205 is: $$ 2205 = 3^2 \times 5 \times 7^2 $$
The prime factorization of 2205 is $3^2 \times 5 \times 7^2$.
More Information
The number 2205 can be expressed as a product of primes, showcasing the fundamental theorem of arithmetic, which states that every integer greater than 1 either is prime itself or can be factored into prime numbers.
Tips
- A common mistake is forgetting to check divisibility for every prime number up to the square root of the number.
- Another mistake is assuming that any odd number is prime without testing for divisibility.
