Final answer:
Alex should cut his two ropes, which measure 15 and 12 feet, into 3-foot lengths, as 3 feet is the greatest common divisor of both rope lengths, allowing equal-sized pieces without any leftover material.
Step-by-step explanation:
Alex has two pieces of rope that measure 15 and 12 feet, and he wants to cut these ropes into equal pieces that are as long as possible.
The greatest length into which both ropes can be evenly cut without any leftover is determined by finding the greatest common divisor (GCD) of the two lengths. The GCD of 15 and 12 is 3 feet. Therefore, Alex should cut the pieces of rope into 3-foot lengths to have pieces that are as long as possible without any waste.
To find the GCD, we can use the Euclidean algorithm or just list the factors of both numbers and find the largest common factor. For 15, the factors are 1, 3, 5, and 15; for 12, the factors are 1, 2, 3, 4, 6, and 12. The largest factor that both numbers share is 3, hence the GCD is 3 feet. If Alex cuts each rope into 3-foot segments, he will get five pieces from the 15-foot rope and four pieces from the 12-foot rope.