Understanding Prime Factorization with an Engaging Perspective

Which number has a prime factorization of three odd numbers and one even number?

• A. 9
• B. 21
• C. 45
• D. 90

Answer: (D)

A number with a prime factorization that includes three odd numbers and one even number must contain both prime factors that are odd and at least one prime factor that is even. The only even prime number is 2. In the case of the number 90, its prime factorization is 2 × 3 × 3 × 5. Here, 2 is the even prime factor, while 3 and 5 are the odd prime factors. Specifically, there are three odd numbers in the factorization: two occurrences of 3 and one occurrence of 5. The other options do not fulfill the requirement of containing one even prime factor and three odd prime factors. For instance, 9 is composed entirely of odd factors (3 × 3), as are 21 (3 × 7) and 45 (3 × 3 × 5). None of these numbers includes an even factor, making 90 the only suitable choice that meets the original condition described in the question.

When you think about numbers and their hidden secrets, prime factorization might just be one of the coolest concepts out there. Ever wondered how to break down a number into its fundamental building blocks? For those preparing for the MTTC 103, let’s dig into a question that's sure to pop up and understand it in style.

So, here’s the question worth pondering: Which number has a prime factorization of three odd numbers and one even number? You’ve got four options: 9, 21, 45, or 90. You might want to grab your thinking cap because this is where the fun begins!

Let’s start the countdown with 90. When you break it down, what do you get? That's right! You land on 2 × 3 × 3 × 5. Here’s the kicker: you've got your even prime factor, which is 2 (the only even prime) and three odd primes—two 3s and one 5. So, bingo! 90 checks all the boxes.

Now, what about the others? Let's give them a glance:

• 9, for instance, is just 3 × 3. All odd, no even factors in sight.

• How about 21? It breaks down to 3 × 7. Yep, still only odds with no even numbers playing in the mix.

• Now, 45? That’s another odd affair with 3 × 3 × 5. Again, no evens.

It’s crazy when you think about it—90's the only number here that plays nicely with the rule of having one even and at least three odd prime factors.

But why does this matter? Understanding prime factorization isn’t just an academic exercise; it helps pave the way in various areas of mathematics and even day-to-day calculations. Look at how math is woven into everything—from cooking (think proportioning ingredients) to budgeting your monthly expenses. It’s all about the numbers!

Prime factorization isn’t just number crunching; it’s about recognizing patterns and relationships between numbers, which can be incredibly powerful. So, the next time you find yourself weighing odd and even primes, remember that’s not just math; that’s a way of seeing the world through the lens of numbers—so logical, so satisfying.

Remember, understanding how to play with numbers gives you a great foundation for more challenging topics in mathematics. Keep that mindset open, engage with numbers, and you'll discover countless opportunities for learning as you prepare for your MTTC 103 journey!