Final answer:
The greatest common divisor (GCD) of 4725 and 3234 is 21, and the least common multiple (LCM) is 226170, found using their canonical representations as products of prime factors.
Step-by-step explanation:
The student is asking to find the greatest common divisor (GCD) and the least common multiple (LCM) of the numbers 4725 and 3234 using the canonical representation. The canonical representation refers to expressing numbers as the product of their prime factors.
Step-by-step to find the GCD and LCM:
First, we find the prime factorization of each number:
4725 = 3³ × 5² × 7
3234 = 2 × 3 × 7 × 7
For GCD, we take the lowest power of all common prime factors:
GCD(4725, 3234) = 3 × 7
For LCM, we take the highest power of all prime factors present in either number:
LCM(4725, 3234) = 2 × 3³ × 5² × 7 × 7
Therefore, the GCD of 4725 and 3234 is 21, and their LCM is 226170.