Question

a. List all possible factorizations of \(4,9,\) and 25 b. What do the factorizations of these three numbers share in common?

Short answer

Answer: The factorizations of 4, 9, and 25 have in common that each number has a factorization that is the square of one of its factors, as they are all perfect squares.

Step-by-step solution

Finding the factorization of 4

To find the factorization of 4, we must find all numbers that, when multiplied together, equal 4. The factor pairs of 4 are (1, 4) and (2, 2). Thus, the factorizations of 4 are 1x4 and 2x2.

Finding the factorization of 9

To find the factorization of 9, we must find all numbers that, when multiplied together, equal 9. The factor pairs of 9 are (1, 9) and (3, 3). Thus, the factorizations of 9 are 1x9 and 3x3.

Finding the factorization of 25

To find the factorization of 25, we must find all numbers that, when multiplied together, equal 25. The factor pairs of 25 are (1, 25) and (5, 5). Thus, the factorizations of 25 are 1x25 and 5x5.

Analyzing the factorizations

Now that we have all the factorizations of the given numbers, we can list them together for easy comparison: - 4: 1x4, 2x2 - 9: 1x9, 3x3 - 25: 1x25, 5x5

Finding common element

Observing these factorizations, we notice that each number has a factorization with a pair of equal factors (i.e., the factorization is a square of one of its factors): - 4: 2x2 - 9: 3x3 - 25: 5x5 So, the factorizations of these three numbers share in common that each number has a factorization that is the square of one of its factors. This is because each number (4, 9, and 25) is a perfect square (an integer that is the square of another integer).