A grocery store gave out to 200 COUPONS customers labeled 1-200 . If a customer receives a coupon

Question

with a prime factor of 200 they win a free turkey. How many winning coupons are there?

Answer 1

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 \( \boxed{120} \).