Final answer:
The greatest common divisor (GCD) of 1000 and 625 is 125, and their least common multiple (LCM) is 5000. It can be verified that the product of the GCD and LCM is equal to the product of the two numbers, affirming the mathematical relationship between GCD and LCM.
Step-by-step explanation:
Finding the GCD and LCM of 1000 and 625
To find the greatest common divisor (GCD) of 1000 and 625, we can use the Euclidean algorithm that involves a sequence of division steps until we reach a remainder of 0. The last non-zero remainder is the GCD. For 1000 and 625:
1000 divided by 625 gives a remainder of 375.
625 divided by 375 gives a remainder of 250.
375 divided by 250 gives a remainder of 125.
250 divided by 125 gives a remainder of 0.
Thus, the GCD is 125.
To find the least common multiple (LCM), we use the formula: LCM(a, b) = (a × b) / GCD(a, b). Using our GCD of 125, we get:
LCM(1000, 625) = (1000 × 625) / 125 = 5000.
Now we'll verify that GCD(1000, 625) × LCM(1000, 625) = 1000 × 625. Indeed, 125 × 5000 = 625000, which is the same as 1000 × 625.