What is the greatest value of the ratio of the least common multiple to the greatest common

What is the greatest value of the ratio of the least common multiple to the greatest common divisor of two natural numbers, if the numbers themselves are related as 39:69?

Decision.
Let the first unknown number x = 39 ∙ k, where the proportionality coefficient k∈ N, then the second number will be y = 69 ∙ k, since it is known from the problem statement that the numbers themselves are related as 39: 69. Let us decompose these numbers into prime factors: x = 39 ∙ k = 3 ∙ 13 ∙ k and y = 69 ∙ k = 3 ∙ 23 ∙ k. Let’s find the smallest common multiple of these numbers LCM (x; y) = 3 ∙ 13 ∙ 23 ∙ k and the greatest common divisor of GCD (x; y) = 3 ∙ k. To determine what the greatest value can be taken by the ratio of the least common multiple to the greatest common divisor of two natural numbers, we find the quotient:
(3 ∙ 13 ∙ 23 ∙ k) / (3 ∙ k) = 13 ∙ 23 = 299.
Answer: The ratio of the least common multiple to the greatest common divisor of two given natural numbers is 299.