Problem 62 - Prealgebra

Question

A radio station is giving away two concert tickets to every sixteenth caller and a dinner for two to every twentieth caller. Which caller will receive both the concert tickets and the dinner? A 32 th B 40 th C 56 th D 80 th

Step-by-Step Solution

Answer

The 80th caller will receive both the concert tickets and the dinner.

Step 1: Understanding the Problem

We need to find the common caller number who receives both concert tickets and a dinner. This means we need to find a number that is a multiple of both 16 and 20.

Step 2: Finding the Least Common Multiple (LCM)

To find the common caller, calculate the least common multiple (LCM) of 16 and 20. We use the highest power of all prime numbers appearing in the factorizations of both numbers: - The prime factorization of 16 is: \(16 = 2^4\).- The prime factorization of 20 is: \(20 = 2^2 \times 5\).The LCM will have the highest powers of all primes present, thus: \[\text{LCM}(16, 20) = 2^4 \times 5 = 80.\]

Step 3: Conclusion

The LCM of 16 and 20 is 80, meaning the caller number 80 will receive both the concert tickets and the dinner.

Key Concepts

Prime Factorization

Prime factorization is a method of expressing a number as a product of its prime numbers. To put it simply, it breaks down a number into the smallest building blocks, which are prime numbers. Understanding prime factorization is crucial for finding the least common multiple (LCM) and solving similar problems where you look for shared traits between numbers.

Consider the number 16. To find its prime factorization, you repeatedly divide the number by the smallest prime number until you reach 1. Here’s how it works:

• 16 is divided by 2 to get 8.
• 8 is divided by 2 to get 4.
• 4 is divided by 2 to get 2.
• 2 is divided by 2 to reach 1.

Thus, the prime factorization of 16 is expressed as \(16 = 2^4\).

Now, let's factorize 20:

• 20 is divided by 2 to get 10.
• 10 is divided by 2 to get 5.
• 5 is a prime number and divides itself to get 1.

So, the prime factorization of 20 is \(20 = 2^2 \times 5\).

These factorizations help us identify the prime numbers involved and their powers, which is essential for finding common measures like the LCM.

Multiples

Multiples are essentially what you end up with when you multiply a number by an integer. They form a sequence that can go on infinitely. Recognizing multiples is key when you’re trying to find commonalities between numbers, as in our exercise with caller awards.

For instance, if you have a number 16, its multiples include:

• 16, 32, 48, 64, and so on. These are calculated by 16 multiplied by integers 1, 2, 3, etc.

Similarly, for the number 20, the multiples are:

• 20, 40, 60, 80, and so forth. These result from 20 multiplied by 1, 2, 3, etc.

Observing multiples helps us in identifying numbers like the least common multiple, which is the smallest number that is a multiple of both integers—16 and 20 in our case—found without them skipping a turn on the call queue.

Common Multiples

Common multiples give us insight into the shared points within multiple sequences of numbers. When you list out the multiples of two numbers, the common multiples are those that appear in both sequences.

In our problem, we're interested in the multiples of 16 and 20. As listed before, their sequences are:

• Multiples of 16: 16, 32, 48, 64, 80, ...
• Multiples of 20: 20, 40, 60, 80, ...

By observing, you can notice that the number 80 appears in both lists. Hence, 80 is a common multiple of 16 and 20.

To find the first (or smallest) common multiple, also known as the least common multiple (LCM), you can use the prime factorizations we've worked out. By choosing the highest powers of all prime factors that appear, you calculate the LCM: \[\text{LCM}(16, 20) = 2^4 \times 5 = 80\]This tells us that the 80th caller fits the conditions of receiving both concert tickets and a dinner. Recognizing common multiples through prime factorization and sequences allows you to solve such problems efficiently.