Question
QUESTION 2 2.1 Express 90 as the product of powers of primes. 2.2 Use prime factors to determine whether 90 a factor of 250.
Ask by Gordon Romero. in South Africa
Feb 24,2025
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Answer
90 can be expressed as \( 2^1 \times 3^2 \times 5^1 \). Since 250's prime factors are \( 2^1 \times 5^3 \), and it lacks the \( 3^2 \) factor, 90 is not a factor of 250.
Solution
Let's solve the problem step by step.
### 2.1 Express 90 as the product of powers of primes.
To express 90 as the product of powers of primes, we need to perform prime factorization.
1. Start with the number 90.
2. Divide by the smallest prime number, which is 2:
\[
90 \div 2 = 45
\]
So, we have \( 90 = 2 \times 45 \).
3. Next, factor 45. The smallest prime number that divides 45 is 3:
\[
45 \div 3 = 15
\]
So, we have \( 45 = 3 \times 15 \), and thus \( 90 = 2 \times 3 \times 15 \).
4. Now, factor 15. The smallest prime number that divides 15 is again 3:
\[
15 \div 3 = 5
\]
So, we have \( 15 = 3 \times 5 \), and thus \( 90 = 2 \times 3 \times 3 \times 5 \).
5. We can express this in terms of powers:
\[
90 = 2^1 \times 3^2 \times 5^1
\]
### 2.2 Use prime factors to determine whether 90 is a factor of 250.
Now, we need to check if 90 is a factor of 250 using its prime factors.
1. First, let's find the prime factorization of 250:
- Start with 250 and divide by the smallest prime number, which is 2:
\[
250 \div 2 = 125
\]
So, we have \( 250 = 2 \times 125 \).
2. Next, factor 125. The smallest prime number that divides 125 is 5:
\[
125 \div 5 = 25
\]
So, we have \( 125 = 5 \times 25 \), and thus \( 250 = 2 \times 5 \times 25 \).
3. Now, factor 25. The smallest prime number that divides 25 is 5:
\[
25 \div 5 = 5
\]
So, we have \( 25 = 5 \times 5 \), and thus \( 250 = 2 \times 5 \times 5 \times 5 \).
4. We can express this in terms of powers:
\[
250 = 2^1 \times 5^3
\]
### Conclusion
Now we can compare the prime factors of 90 and 250:
- The prime factorization of 90 is \( 2^1 \times 3^2 \times 5^1 \).
- The prime factorization of 250 is \( 2^1 \times 5^3 \).
To determine if 90 is a factor of 250, we check if all the prime factors of 90 are present in 250 with at least the same or greater powers:
- For \( 2^1 \): Present in both.
- For \( 3^2 \): Not present in 250.
- For \( 5^1 \): Present in both.
Since \( 3^2 \) is not present in the factorization of 250, we conclude that **90 is not a factor of 250**.
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Simplify this solution Extra Insights
To express 90 as the product of powers of primes, we start with the prime factorization. 90 can be broken down into its prime factors: 90 = 2^1 × 3^2 × 5^1. So, there you have it, 90 is the prime power product of these numbers! Now, to determine if 90 is a factor of 250, we first find the prime factorization of both numbers. The prime factorization of 250 is 2^1 × 5^3. Since 90 has a 3 in its prime factorization (3^2), and 250 does not include a factor of 3, we can confidently say that 90 is not a factor of 250!
