Question
You pick a card at random.
3
4
5
What is P(factor of 56 or divisor of 90)?
Write your answer as a percentage.
3
4
5
What is P(factor of 56 or divisor of 90)?
Write your answer as a percentage.
Answers
GPT-4o mini
To find the probability \( P(\text{factor of 56 or divisor of 90}) \), we first need to identify the factors of 56 and the divisors of 90.
1. **Factors of 56**:
- The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.
2. **Divisors of 90**:
- The divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Now, we need to find all the numbers that are either a factor of 56 or a divisor of 90, then count the distinct numbers.
### Factors of 56:
- **1**, **2**, **4**, **7**, **8**, **14**, **28**, **56**
### Divisors of 90:
- **1**, **2**, **3**, **5**, **6**, **9**, **10**, **15**, **18**, **30**, **45**, **90**
### Union of Sets (Factors of 56 or Divisors of 90):
Combining the two lists:
- From factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
- From divisors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Now, let's combine and list the distinct values:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 18, 28, 30, 45, 56, 90
### Count Distinct Values:
The distinct numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 18, 28, 30, 45, 56, 90.
This gives us a total of 18 distinct numbers.
### Total Possible Outcomes:
There are 3 possible cards to pick from: 3, 4, or 5.
### Probability Calculation:
The numbers we have counted (1 to 90) do not affect the 3 cards picked, so we consider how many of these numbers are among the cards.
- The cards available are: 3, 4, and 5.
- Among these, the factor of 56 or divisor of 90 is:
- 3: Divisor of 90
- 4: Factor of 56
- 5: Divisor of 90
Each card falls in our earlier union.
Thus, all three cards are either a factor of 56 or a divisor of 90.
### Final Probability:
The probability is:
\[
P(\text{factor of 56 or divisor of 90}) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}} = \frac{3}{3} = 1
\]
Converting this into a percentage:
\[
P = 1 \times 100\% = 100\%
\]
So, the probability \( P(\text{factor of 56 or divisor of 90}) \) is **100%**.
1. **Factors of 56**:
- The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56.
2. **Divisors of 90**:
- The divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90.
Now, we need to find all the numbers that are either a factor of 56 or a divisor of 90, then count the distinct numbers.
### Factors of 56:
- **1**, **2**, **4**, **7**, **8**, **14**, **28**, **56**
### Divisors of 90:
- **1**, **2**, **3**, **5**, **6**, **9**, **10**, **15**, **18**, **30**, **45**, **90**
### Union of Sets (Factors of 56 or Divisors of 90):
Combining the two lists:
- From factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
- From divisors of 90: 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90
Now, let's combine and list the distinct values:
- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 18, 28, 30, 45, 56, 90
### Count Distinct Values:
The distinct numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 18, 28, 30, 45, 56, 90.
This gives us a total of 18 distinct numbers.
### Total Possible Outcomes:
There are 3 possible cards to pick from: 3, 4, or 5.
### Probability Calculation:
The numbers we have counted (1 to 90) do not affect the 3 cards picked, so we consider how many of these numbers are among the cards.
- The cards available are: 3, 4, and 5.
- Among these, the factor of 56 or divisor of 90 is:
- 3: Divisor of 90
- 4: Factor of 56
- 5: Divisor of 90
Each card falls in our earlier union.
Thus, all three cards are either a factor of 56 or a divisor of 90.
### Final Probability:
The probability is:
\[
P(\text{factor of 56 or divisor of 90}) = \frac{\text{number of favorable outcomes}}{\text{total outcomes}} = \frac{3}{3} = 1
\]
Converting this into a percentage:
\[
P = 1 \times 100\% = 100\%
\]
So, the probability \( P(\text{factor of 56 or divisor of 90}) \) is **100%**.