Exploring Pairs of Numbers with GCF of 30 and LCM of 3150
In this article, we'll delve into the fascinating world of number theory, specifically focusing on pairs of numbers that share a greatest common factor (GCF) of 30 and a least common multiple (LCM) of 3150. We will explore the mathematical principles and techniques involved in identifying and understanding these pairs. Let's begin!
Introduction to GCF and LCM
The greatest common factor (GCF) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. On the other hand, the least common multiple (LCM) is the smallest positive integer that is divisible by both numbers without leaving a remainder. These concepts play a crucial role in various areas of mathematics, including number theory, algebra, and cryptography.
Problem Setup
The problem at hand is to find all pairs of numbers such that their GCF is 30 and their LCM is 3150. Let's denote this pair of numbers as 30a and 30b, where a and b are integers that do not share any common factors other than 1 (since their GCF is 30).
Relationship Between GCF and LCM
There is a well-known relationship between GCF and LCM of two numbers, which is given by:
[text{GCF}(30a, 30b) times text{LCM}(30a, 30b) 30a times 30b]Given that the GCF is 30 and the LCM is 3150, we can substitute these values into the equation:
[30 times 3150 30a times 30b]This simplifies to:
[3150 a times b]Since the GCF of 30a and 30b is 30, a and b must be coprime (i.e., their greatest common factor is 1). Therefore, the product a * b must be equal to the product of the numbers divided by the square of their GCF:
[a times b frac{3150}{30^2} frac{3150}{900} 3.5]However, since a and b must be integers, let's correct the calculation:
[a times b frac{3150}{30} 105]Thus, we need to find all pairs of coprime integers (a, b) such that their product is 105.
Determining Coprime Pairs
To find the pairs of coprime integers whose product is 105, we first need to factorize 105. The prime factorization of 105 is:
[105 3 times 5 times 7]Given that a and b are coprime, the only pairs of numbers (a, b) that satisfy this condition are:
[1 times 105, quad 3 times 35, quad 5 times 21, quad 7 times 15]However, since a and b are factors of 105, we must ensure they are coprime. The correct pairs are:
[1 times 105, quad 3 times 35, quad 5 times 21, quad 7 times 15]Therefore, the valid pairs are:
[begin{aligned} (30 times 1, 30 times 105) (30, 3150) (30 times 3, 30 times 35) (90, 1050) (30 times 5, 30 times 21) (150, 630) (30 times 7, 30 times 15) (210, 450)end{aligned}]Thus, the pairs of numbers that have a GCF of 30 and an LCM of 3150 are (30, 3150), (90, 1050), (150, 630), and (210, 450).
Conclusion
In conclusion, we have successfully identified all pairs of numbers that have a GCF of 30 and an LCM of 3150. The pairs are (30, 3150), (90, 1050), (150, 630), and (210, 450). By understanding the relationship between GCF and LCM, we can solve similar problems involving different numbers.