lenge Aaron rolls a die. Every time he rolls a prime number, he gets 5 points. Every time he does not roll a prime number, he loses 3 points. At the end of the game, his score is 0 . What is the least number of rolls he could have made? (Hint: The points he won and lost must be the same.)

To solve this, let's define some clear variables. Let \( p \) be the number of times Aaron rolls a prime number, and \( n \) be the number of times he rolls a non-prime number. The prime numbers on a standard die (1 to 6) are 2, 3, and 5, which account for 3 possible outcomes. The non-prime numbers are 1, 4, and 6, totaling 3 as well. For each prime number rolled, Aaron earns 5 points, leading to a total score from prime rolls of \( 5p \). For each non-prime roll, he loses 3 points, which totals \( -3n \). At the end, his score is 0, so we set up the equation: \[ 5p - 3n = 0 \] This can be rearranged to: \[ 5p = 3n \] To find the least number of rolls, we can express \( n \) in terms of \( p \): \[ n = \frac{5p}{3} \] Since \( n \) must be a whole number, \( 5p \) has to be divisible by 3. This means \( p \) must be a multiple of 3 for \( n \) to be an integer. The smallest \( p \) that meets this requirement is \( p = 3 \). Substituting \( p = 3 \) back into the equation for \( n \): \[ n = \frac{5(3)}{3} = 5 \] Now, the total number of rolls \( r \) can be calculated as: \[ r = p + n = 3 + 5 = 8 \] So, the least number of rolls Aaron could have made is \( \boxed{8} \).