Brandon has 32 stamps. He wants to display the stamps in rows, with the same number of stamps in each row. How many different ways can he display the stamps? Explain.

To determine how many different ways Brandon can display his 32 stamps in rows with the same number of stamps in each row, we need to find the divisors of 32. Each divisor represents a different way to arrange the stamps. ### Step 1: Find the divisors of 32 First, we can find the prime factorization of 32: \[ 32 = 2^5 \] ### Step 2: Calculate the number of divisors To find the number of divisors of a number based on its prime factorization, we can use the formula: \[ \text{Number of divisors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \] where \(e_1, e_2, e_3, \ldots\) are the exponents in the prime factorization. For \(32 = 2^5\): - The exponent \(e_1 = 5\). Thus, the number of divisors is: \[ (5 + 1) = 6 \] ### Step 3: List the divisors Now, let's list the divisors of 32: 1. \(1\) (1 row of 32 stamps) 2. \(2\) (2 rows of 16 stamps) 3. \(4\) (4 rows of 8 stamps) 4. \(8\) (8 rows of 4 stamps) 5. \(16\) (16 rows of 2 stamps) 6. \(32\) (32 rows of 1 stamp) ### Conclusion Brandon can display his 32 stamps in **6 different ways**. Each way corresponds to a different number of rows, with the same number of stamps in each row.