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Chapter 5: Problem 87

Find the product of the greatest common divisor of 24 and 27 and the least common multiple of 24 and 27. Compare this result to the product of 24 and 27 . Write a conjecture based on your observation.

Short Answer

Expert verified
GCD of 24 and 27 is 3, LCM is 216. The product of the GCD and LCM (648) is equal to the product of 24 and 27 (648). Therefore, a potential conjecture based on this result could be: the product of the GCD and the LCM of two numbers is equal to the product of these numbers.

Step by step solution

01

Determine the GCD of 24 and 27

The first step is to determine the greatest common divisor of the two numbers, 24 and 27. Use the Euclidean algorithm to compute this. The prime factors of 24 are 2, 2, 2, 3 (or \(2^3 \cdot 3\)) and the prime factors of 27 are 3, 3, 3 (or \(3^3\)). The GCD is the product of the smallest power of each common prime factor in the numbers. Here, the common factor is 3 and the smallest power is 1. Therefore, the GCD of 24 and 27 is 3.
02

Determine the LCM of 24 and 27

The next step is to determine the least common multiple of 24 and 27. The LCM is the product of the greatest power of all prime factors that appear in the numbers. That is \(2^3 \cdot 3^3\), which is 216.
03

Compute the product of GCD and LCM

Compute the product of the greatest common divisor and the least common multiple. \(3 \times 216 = 648\)
04

Compare with the product of 24 and 27

Find the product of 24 and 27. \(24 \times 27 = 648\)
05

Write the conjecture

Observe the results and formulate a conjecture. In this case, the conjecture could be that the product of the least common multiple and greatest common divisor of any two numbers is equal to the product of the numbers themselves. However, this needs further validation for other combinations of numbers.

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