What number has the prime factorization of
?
The Correct Answer and Explanation is:
The correct answer is 100.
To solve this problem, we are asked to identify the number with the prime factorization 22⋅522^2 \cdot 5^2. Prime factorization expresses a number as a product of its prime factors raised to certain powers. Here, the primes involved are 2 and 5.
Start by evaluating each component:
- 22=2×2=42^2 = 2 \times 2 = 4
- 52=5×5=255^2 = 5 \times 5 = 25
Then multiply the results:
- 4×25=1004 \times 25 = 100
Thus, the number with the prime factorization 22⋅522^2 \cdot 5^2 is 100.
Now, let’s elaborate on why this method works and what it teaches us. Prime factorization is foundational in number theory. Every whole number greater than 1 can be uniquely expressed as a product of prime numbers. This is called the Fundamental Theorem of Arithmetic. The process is useful in various areas of mathematics, including simplifying fractions, finding least common multiples, and solving problems involving divisibility.
The given prime factorization 22⋅522^2 \cdot 5^2 tells us that the number must be divisible by both 4 and 25, because those values are powers of primes. Multiplying them together gives the original number. This approach not only helps identify the number, but it also builds an understanding of how numbers are constructed from their prime components.
Understanding the concept of exponentiation is also critical. Exponents represent repeated multiplication. For example, 222^2 means using two 2s in a product, which results in 4. Similarly, 525^2 results in 25. Recognizing and using these patterns makes calculations efficient and manageable.
In this case, the final product, 100, represents a simple but powerful demonstration of how prime numbers combine to form composite numbers.
