Question
A grocery store gave out to 200 COUPONS customers labeled 1-200 . If a customer receives a coupon
with a prime factor of 200 they win a free turkey. How many winning coupons are there?
with a prime factor of 200 they win a free turkey. How many winning coupons are there?
Answers
GPT-4o mini
To determine how many winning coupons a customer can receive, we first need to identify the prime factors of 200.
1. **Finding the prime factorization of 200:**
\[
200 = 2^3 \times 5^2
\]
From this factorization, we see that the prime factors of 200 are 2 and 5.
2. **Identifying winning coupons:**
A customer will win a free turkey if their coupon number, which ranges from 1 to 200, has either 2 or 5 as a prime factor. In other words, we need to count the numbers from 1 to 200 that are divisible by either 2 or 5.
3. **Using the principle of inclusion-exclusion:**
- Let \( A \) be the set of numbers from 1 to 200 that are divisible by 2.
- Let \( B \) be the set of numbers from 1 to 200 that are divisible by 5.
We need to find \( |A \cup B| \), which can be calculated using the formula:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
4. **Calculating \(|A|\) (divisible by 2):**
The numbers divisible by 2 from 1 to 200 are \( 2, 4, 6, \ldots, 200 \). This is an arithmetic sequence with:
- First term \( a = 2 \)
- Last term \( l = 200 \)
- Common difference \( d = 2 \)
The number of terms \( n \) can be found using:
\[
n = \frac{l - a}{d} + 1 = \frac{200 - 2}{2} + 1 = 100
\]
So \( |A| = 100 \).
5. **Calculating \(|B|\) (divisible by 5):**
The numbers divisible by 5 from 1 to 200 are \( 5, 10, 15, \ldots, 200 \). This is an arithmetic sequence with:
- First term \( a = 5 \)
- Last term \( l = 200 \)
- Common difference \( d = 5 \)
The number of terms \( n \) can be found using:
\[
n = \frac{l - a}{d} + 1 = \frac{200 - 5}{5} + 1 = 40
\]
So \( |B| = 40 \).
6. **Calculating \(|A \cap B|\) (divisible by both 2 and 5, i.e., divisible by 10):**
The numbers divisible by 10 from 1 to 200 are \( 10, 20, 30, \ldots, 200 \). This is an arithmetic sequence with:
- First term \( a = 10 \)
- Last term \( l = 200 \)
- Common difference \( d = 10 \)
The number of terms \( n \) can be found using:
\[
n = \frac{l - a}{d} + 1 = \frac{200 - 10}{10} + 1 = 20
\]
So \( |A \cap B| = 20 \).
7. **Using inclusion-exclusion to find \( |A \cup B| \):**
\[
|A \cup B| = |A| + |B| - |A \cap B| = 100 + 40 - 20 = 120
\]
Thus, the total number of winning coupons is \( \boxed{120} \).
1. **Finding the prime factorization of 200:**
\[
200 = 2^3 \times 5^2
\]
From this factorization, we see that the prime factors of 200 are 2 and 5.
2. **Identifying winning coupons:**
A customer will win a free turkey if their coupon number, which ranges from 1 to 200, has either 2 or 5 as a prime factor. In other words, we need to count the numbers from 1 to 200 that are divisible by either 2 or 5.
3. **Using the principle of inclusion-exclusion:**
- Let \( A \) be the set of numbers from 1 to 200 that are divisible by 2.
- Let \( B \) be the set of numbers from 1 to 200 that are divisible by 5.
We need to find \( |A \cup B| \), which can be calculated using the formula:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
4. **Calculating \(|A|\) (divisible by 2):**
The numbers divisible by 2 from 1 to 200 are \( 2, 4, 6, \ldots, 200 \). This is an arithmetic sequence with:
- First term \( a = 2 \)
- Last term \( l = 200 \)
- Common difference \( d = 2 \)
The number of terms \( n \) can be found using:
\[
n = \frac{l - a}{d} + 1 = \frac{200 - 2}{2} + 1 = 100
\]
So \( |A| = 100 \).
5. **Calculating \(|B|\) (divisible by 5):**
The numbers divisible by 5 from 1 to 200 are \( 5, 10, 15, \ldots, 200 \). This is an arithmetic sequence with:
- First term \( a = 5 \)
- Last term \( l = 200 \)
- Common difference \( d = 5 \)
The number of terms \( n \) can be found using:
\[
n = \frac{l - a}{d} + 1 = \frac{200 - 5}{5} + 1 = 40
\]
So \( |B| = 40 \).
6. **Calculating \(|A \cap B|\) (divisible by both 2 and 5, i.e., divisible by 10):**
The numbers divisible by 10 from 1 to 200 are \( 10, 20, 30, \ldots, 200 \). This is an arithmetic sequence with:
- First term \( a = 10 \)
- Last term \( l = 200 \)
- Common difference \( d = 10 \)
The number of terms \( n \) can be found using:
\[
n = \frac{l - a}{d} + 1 = \frac{200 - 10}{10} + 1 = 20
\]
So \( |A \cap B| = 20 \).
7. **Using inclusion-exclusion to find \( |A \cup B| \):**
\[
|A \cup B| = |A| + |B| - |A \cap B| = 100 + 40 - 20 = 120
\]
Thus, the total number of winning coupons is \( \boxed{120} \).