One table at a a bake sale has 150 mini cupcakes. Another table has 180 mini cookies. Which table allows for more square arrangements when all the mini cupcakes and mini cookies are displayed?

Answer 1

To determine which table allows for more square arrangements, we need to find the largest perfect square that can divide the number of mini cupcakes and mini cookies.

1. Mini Cupcakes: 150

   • Find the prime factorization of 150: \[ 150 = 2 \times 3 \times 5^2 \]
   • The square factors can be obtained by taking the minimum exponent for each prime factor to create a perfect square:
     • The exponent of 2 is 1 (take 0),
     • The exponent of 3 is 1 (take 0),
     • The exponent of 5 is 2 (take 2).
   • Therefore, the largest perfect square factor of 150 is: \[ 5^2 = 25 \]

2. Mini Cookies: 180

   • Find the prime factorization of 180: \[ 180 = 2^2 \times 3^2 \times 5 \]
   • Again, we take the minimum exponent for each prime factor:
     • The exponent of 2 is 2 (take 2),
     • The exponent of 3 is 2 (take 2),
     • The exponent of 5 is 1 (take 0).
   • Thus, the largest perfect square factor of 180 is: \[ 2^2 \times 3^2 = 4 \times 9 = 36 \]

3. Comparison:

   • The largest perfect square factor for the mini cupcakes (150) is 25.
   • The largest perfect square factor for the mini cookies (180) is 36.

Since 36 (for the mini cookies) is greater than 25 (for the mini cupcakes), the table with 180 mini cookies allows for more square arrangements.