Final answer:
The greatest number that divides both 204 and 64 without leaving a remainder is 16, which is found by listing the factors of each number and identifying the highest one they share.
Step-by-step explanation:
The greatest number that divides 204 and 64 without leaving a remainder is sought after, which is a concept known as finding the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) in mathematics. To find this, we can use the Euclidean algorithm or simply list the factors of each number and determine the highest one they have in common.
Factors of 204: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 204
Factors of 64: 1, 2, 4, 8, 16, 32, 64
From the lists, the highest common factor of 204 and 64 is 16, which is also the greatest number that divides both without leaving a remainder.