A toddler has 36 blocks of which 16 are blue and 20 are pink they want to sort the block so that each group has both colors of blocks the toddler also wants each group to have an equal number of blue blocks and an equal number of pink blocks which answer best shows how the toddlers should sort the blocks using the GCF and the distributive property of multiplication over addition

To sort the blocks into groups that have an equal number of blue and pink blocks, we can utilize the greatest common factor (GCF) approach. We begin by determining the GCF of the number of blue blocks (16) and pink blocks (20).

**Step 1: Find the GCF of 16 and 20.**

The factors of 16 are:
1, 2, 4, 8, 16

The factors of 20 are:
1, 2, 4, 5, 10, 20

The common factors of 16 and 20 are:
1, 2, 4

So, the GCF is **4**.

**Step 2: Determine how many groups can be formed.**

If each group can have 4 blocks of each color, we divide the total number of each color by the GCF.

- For blue blocks:
\( \frac{16 \text{ blue blocks}}{4} = 4 \) groups

- For pink blocks:
\( \frac{20 \text{ pink blocks}}{4} = 5 \) groups

**Step 3: Understanding the group sizes.**

Since we want each group to have the same number of blue and pink blocks and to sort in a balanced manner, we'll set up the groups as follows:

- Each group will have **4 blue blocks** and **4 pink blocks**.

**Step 4: Using the distributive property.**

We can express the total number of blocks in terms of groups. If we denote the number of groups as \( g \), then:

- Total number of blue blocks: \( 4g = 16 \)
- Total number of pink blocks: \( 4g = 20 \)

Where \( g \) is the number of groups.

From \( 4g = 16 \): \( g = \frac{16}{4} = 4 \) groups of blue blocks.

From \( 4g = 20 \): \( g = \frac{20}{4} = 5 \) groups of pink blocks.

**Conclusion:**

To satisfy the toddler's sorting requirement, the best way to group the blocks would be to have \(4\) groups of \(4\) blue blocks and \(5\) groups of \(4\) pink blocks. However, as we need both colors in the same groups, the following distribution works:

- **4 groups** total will be formed, each containing:

- **4 blue blocks** and **4 pink blocks**.

In total, that’s \(4 \times (4 \text{ blue} + 4 \text{ pink}) = 32 \text{ blocks (16 blue + 16 pink)}\). The remaining pink blocks can be kept as an extra group that deters from the even distribution but follows the toddler’s initial intent of sorting.

In summary, the answer shows that the toddler should ideally sort the blocks into **4 groups**, each containing **4 blue blocks** and **4 pink blocks**. The toddler can use the 4 remaining pink blocks in excess for other group sorting, in alignment with appropriate mathematical principles.