To find the smallest positive integer that is a multiple of both 24 and 45, we can find the least common multiple (LCM) of the two numbers.
Prime factorizing 24, we get: 2^3 x 3^1
Prime factorizing 45, we get: 3^2 x 5^1
To find the LCM, we take the highest power of each prime factor that appears in either factorization:
2^3 x 3^2 x 5^1 = 360
Therefore, the smallest positive integer that is a multiple of both 24 and 45 is 360.