2025 AMC 8 Problem 22

Factoring by Grouping Method

Express the total number of hooks algebraically, then add a constant to both sides so the equation can be factored into the product of two integers.

Step 1: Define the variables and identify the number of gaps

Let $c$ be the number of coats and $e$ be the number of empty hooks in each equally spaced gap. To find the total number of gaps, we count the gap before the first coat, the gaps between the coats, and the gap after the last coat. For $c$ coats, there are $c-1$ gaps between them, plus the $2$ gaps at the ends.

Step 2: Set up the equation for the total number of hooks

The total number of hooks on the rack is the sum of the hooks holding coats and the empty hooks. We are given that the total is $35$.

Step 3: Factor the equation

To solve this equation with two integer variables, we want to express it as a product of factors. Notice the term $(c+1)e$. If we add $1$ to both sides of our equation, we can group the terms to create a common factor of $(c+1)$.

Step 4: Establish minimum bounds based on the problem conditions

The problem states there is at least $1$ coat, meaning $c\ge 1$. It also states there is at least $1$ empty hook overall. Since all $c+1$ gaps must have the same number of empty hooks, each gap must have at least $1$ empty hook, meaning $e\ge 1$. We apply these inequalities to our factors.

Step 5: Find the valid factors of 36

We need to find all integer factor pairs of $36$ where both factors are at least $2$. The full list of factors of $36$ is $1,2,3,4,6,9,12,18,36$. We exclude $1$ and $36$ because neither factor is allowed to be $1$. The remaining $7$ factors represent the valid values for $(c+1)$.

Algebraic Division Shortcut

Instead of factoring, solve the total hook equation for the number of empty hooks, and use divisibility rules to see when the result is an integer.

Step 1: Solve the hook equation for the empty hooks

Using the same initial equation for the total number of hooks, we can isolate $e$ (the number of empty hooks per gap). Since $e$ must be an integer, this turns the problem into a divisibility check.

Step 2: Rewrite the fraction to reveal the factors

To easily see when $e$ is an integer, we can manipulate the numerator so that it includes a multiple of the denominator $(c+1)$. We subtract and add $1$ in the numerator.

Step 3: Determine valid values for the number of coats

For $e$ to be an integer, $c+1$ must divide perfectly into $36$. Just as before, $c\ge 1$ requires $c+1\ge 2$. Also, $e\ge 1$ requires that the fraction evaluate to at least $2$, meaning $c+1\le 18$. The divisors of $36$ strictly between $1$ and $36$ give us exactly $7$ valid values for $c+1$.

Phantom Object Method (Periodic Grouping)

By adding one imaginary coat to the right end of the rack, the entire arrangement becomes a repeating sequence of identical blocks, turning the problem into simple division.

Step 1: Visualize the repeating pattern of hooks

Let $e$ be the number of empty hooks in each gap. The rack starts with $e$ empty hooks, followed by $1$ coat. This pattern repeats for every coat. However, after the final coat, there is one last gap of $e$ empty hooks that does not have a coat after it.

Step 2: Introduce a phantom coat to complete the pattern

Imagine placing one extra, imaginary ‘phantom’ coat at the very right end of the rack. Now, every single gap of empty hooks is perfectly paired with a coat immediately to its right. The entire rack is made of identical repeating blocks.

Step 3: Relate the blocks to the total number of items

If there are $c$ real coats, adding the phantom coat means there are $c+1$ identical blocks in total. The total number of hooks, including our phantom one, is $35+1=36$. The total items must equal the number of blocks multiplied by the size of each block.

Step 4: Count the valid configurations

Because there is at least one real coat, the number of blocks $(c+1)$ must be at least $2$. Because there is at least one empty hook total, the block size $(e+1)$ must also be at least $2$. We are looking for all integer factors of $36$ that are between $2$ and $18$ inclusive.