?\ 3 \times 108\ 3 \times 36\ 3 \times 12\ 3 \times 4\ 2 \times 2
The Correct Answer and Explanation is:
The correct answer is 324.
To find the unknown number at the top of the factor tree, multiply all the factors together: 3 × 108 = 324 This can be further verified by breaking down the factor tree: 108 = 3 × 36 36 = 3 × 12 12 = 3 × 4 4 = 2 × 2
Therefore, the full prime factorization is: 3 × 3 × 3 × 3 × 2 × 2 = 324
Explanation: A factor tree is a visual method for breaking down a composite number into its prime components. In this problem, each stage of the factor tree decomposes a number into smaller parts until only prime numbers remain. Starting from 324, we observe that it is divisible by 3 repeatedly until we are left with 4, which in turn is broken into 2 and 2.
The prime numbers used in this tree are 2 and 3. This means the prime factorization of 324 is: 3⁴ × 2²
This factorization can be verified by calculating: 3 × 3 = 9 9 × 3 = 27 27 × 3 = 81 81 × 2 = 162 162 × 2 = 324
Prime factorization has many applications in mathematics. It is essential when simplifying fractions, finding the greatest common divisor (GCD), and least common multiple (LCM). In algebra, it aids in solving polynomial identities and factoring expressions.
This method is systematic and works well with any composite number. By starting with the smallest prime factor and continuously dividing, one ensures all resulting numbers are indeed prime. The unique prime factorization theorem guarantees that every composite number has one and only one prime factorization, regardless of the order in which the factors are written.
Thus, 324 is the original number, and its factor tree illustrates the efficiency and reliability of prime decomposition.
