The question asks:

"Is the number of factor pairs for a number equal to the number of arrays for the same number? Explain."

Understanding the Question:

1. Factor pairs: A factor pair of a number $N$ consists of two numbers $(a, b)$ such that $a \times b = N$.

   • Example: For $N = 12$, the factor pairs are: $(1, 12), (2, 6), (3, 4)$

2. Arrays: Here, "arrays" refers to rectangular arrangements of objects (using rows and columns) where the total number of objects equals $N$.
   Each array corresponds to a factor pair:

   • Rows = one factor
   • Columns = the other factor

   For $N = 12$, the arrays would look like:

   • 1 row of 12 (1 × 12)
   • 2 rows of 6 (2 × 6)
   • 3 rows of 4 (3 × 4)

Answer:

Yes, the number of factor pairs for a number is equal to the number of arrays for that number.

Explanation:

Each array corresponds directly to one factor pair:

• In an array with $r$ rows and $c$ columns, $r \times c = N$.
• This relationship matches exactly with the factor pairs of $N$.

Thus, for any number $N$:

• The number of factor pairs equals the number of possible arrays because both represent ways of organizing the product $r \times c = N$.

Example to Illustrate:

Let $N = 12$:

• Factor pairs: $(1, 12), (2, 6), (3, 4)$ → 3 factor pairs.
• Arrays: $1 \times 12, 2 \times 6, 3 \times 4$ → 3 arrays.

Both cases result in 3 possibilities.

If you have further questions or need clarification, let me know! 😊

Related Questions:

1. How do you calculate all the factor pairs of a number systematically?
2. How can you visualize arrays for a given number?
3. What is the relationship between factors and divisors?
4. Does a perfect square number have an odd or even number of factor pairs? Why?
5. How can arrays help in understanding multiplication and factors?

Tip:

To find the total number of factor pairs of a number, count its divisors and group them into pairs. For perfect squares, one factor will repeat.