a. Two of the given numbers have square arrays. Make a conjecture about the number of factors for square numbers.


A. The number of factors is a prime number.
B. The number of factors is a square number.
C. The number of factors is an even number.
D. The number of factors is an odd number.

Sketch as many different rectangular (including square) arrays as possible for each of the following numbers: 15, 81, 30, 25, 17. Use these arrays for the following:


Part 1:
a. Two of the given numbers have square arrays. Make a conjecture about the number of factors for square numbers.

A. The number of factors is a prime number.
B. The number of factors is a square number.
C. The number of factors is an even number.
D. The number of factors is an odd number.

Each rectangular array of squares gives information about the number of factors of a number. Two rectangles can be formed for the number 6, showing that 6 has factors of 2, 3, 1, and 6.


4159

Sketch as many different rectangular (including square) arrays as possible for each of the following numbers: 15, 81, 30, 25, 17. Use these arrays for the following:

Part 1:
a. Two of the given numbers have square arrays. Make a conjecture about the number of factors for square numbers.

A. The number of factors is a prime number.
B. The number of factors is a square number.
C. The number of factors is an even number.
D. The number of factors is an odd number.


Explain how your sketches support your conjecture.

A. There is one square array and the rest are nonsquare rectangular arrays.
B. There is one nonsquare rectangular array and the rest are square.
C. There are only square arrays.
D. There are only nonsquare rectangular arrays.
E. There are an even number of arrays.

so, what have you done so far?

To make a conjecture about the number of factors for square numbers, we first need to understand what factors are.

Factors are the numbers that can be multiplied together to give a certain number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, since these numbers can be multiplied to give 12.

Now, let's consider square numbers. A square number is the result of multiplying a number by itself. For example, 4 is a square number because 4 * 4 = 16.

Based on this information, we can make a conjecture about the number of factors for square numbers.

Conjecture: The number of factors for square numbers is an odd number.

To test this conjecture, you can choose several square numbers, such as 4, 9, 16, and 25. Calculate the factors of each square number and observe the pattern. You will find that the number of factors for each square number is indeed an odd number.

For example, taking the square number 4, the factors are 1, 2, and 4. There are three factors, which is an odd number.

Therefore, the correct answer to your question is (D) The number of factors is an odd number.