Introduction

When dealing with the greatest common divisor (GCD) and least common multiple (LCM) of two numbers, it is often valuable to understand how these concepts can be used to derive possible values of those numbers. This article focuses on solving the problem where (text{gcd}(m, n) 18) and (text{lcm}(m, n) 720) to find all possible pairs of (m) and (n).

Understanding the Relationship Between GCD and LCM

The relationship between the GCD and LCM of two numbers can be expressed with the equation:

[text{gcd}(m, n) times text{lcm}(m, n) m times n]

Given the problem, we know that:

[text{gcd}(m, n) 18]

[text{lcm}(m, n) 720]

Substituting these values into the equation, we get:

[18 times 720 m times n]

Calculating the left side:

[18 times 720 12960]

Thus, we have:

[m times n 12960]

Expressing m and n in Terms of a and b

Given that (text{gcd}(m, n) 18), we can express (m) and (n) in terms of (a) and (b) as follows:

[m 18a]

[n 18b]

where (text{gcd}(a, b) 1).

Deriving Possible Values of a and b

Substituting these expressions into the product equation:

[18a times 18b 12960]

This simplifies to:

[324ab 12960]

Dividing both sides by 324:

[ab frac{12960}{324} 40]

Now we need to find pairs ((a, b)) such that (ab 40) and (text{gcd}(a, b) 1). The factor pairs of 40 are:

(1 times 40) (2 times 20) (4 times 10) (5 times 8)

Among these pairs, we need to check which pairs are coprime (i.e., gcd(a, b) 1):

(1 times 40): (text{gcd}(1, 40) 1)

(2 times 20): (text{gcd}(2, 20) 2)

(4 times 10): (text{gcd}(4, 10) 2)

(5 times 8): (text{gcd}(5, 8) 1)

The coprime pairs are ((1, 40)) and ((5, 8)).

Determining Possible Values of m and n

Now we can find the values of (m) and (n) based on these coprime pairs:

For ((a, b) (1, 40)):

(m 18 times 1 18) (n 18 times 40 720)

For ((a, b) (5, 8)):

(m 18 times 5 90) (n 18 times 8 144)

Thus, the possible pairs of (m) and (n) are:

((18, 720)) ((720, 18)) ((90, 144)) ((144, 90))

Conclusion

In summary, the pairs ((m, n)) that satisfy (text{gcd}(m, n) 18) and (text{lcm}(m, n) 720) are:

((18, 720)) ((720, 18)) ((90, 144)) ((144, 90))

This problem demonstrates the application of number theory in solving equations involving GCD and LCM and highlights the importance of using these concepts to derive possible values efficiently.