Two of the given numbers have square arrays. Make a coniecture about the number of factors for square numbers. Explain how your sketches support your conjecture.
Based on the given information, we are told that two of the given numbers have square arrays. This means that two of the numbers can be expressed as the product of two equal factors. Let's call these two numbers A and B.
Now, let's consider the factors of A. Since A can be expressed as the product of two equal factors, it means that A has an odd number of factors. This is because any factor of A can be divided into two equal factors (except for one factor which is the square root of A). For example, if A = 36, its factors are 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors in total).
Similarly, let's consider the factors of B. Just like A, B can also be expressed as the product of two equal factors, meaning B also has an odd number of factors. For instance, if B = 16, its factors are 1, 2, 4, 8, 16 (5 factors in total).
From these observations, we can make the following conjecture: "If a number can be expressed as the product of two equal factors, then it will have an odd number of factors."
To support this conjecture further, let's consider some more examples. Take the number 64, which has the factors 1, 2, 4, 8, 16, 32, 64. Again, we see that 64 has 7 factors, an odd number. Similarly, the number 100 has factors 1, 2, 4, 5, 10, 20, 25, 50, 100, totaling 9 factors, which is also odd.
These examples support our conjecture since all the numbers that can be expressed as the product of two equal factors have an odd number of factors. However, it is important to note that not all numbers with an odd number of factors can be expressed as the product of two equal factors (e.g., 15, which has factors 1, 3, 5, 15).