Question

How do you make a list of possible pairs of numbers that have a $LCM$ of $48$?

Answer 1

Hint:This question is related to Least common multiple of two numbers. In this question we need to find the list of possible pairs of numbers which have $LCM$ equal to $48$. To solve this question we first find all factors of $48$ and pair them with $48$ to get a list of possible pairs.

Complete step by step solution:
Let us try to solve this question in which we are asked to find the list of possible pairs of numbers which have $LCM$ of $48$. Before moving to solve this question we need to know the definition of factors of a number, greatest common divisor of two numbers. Factor of a number $n$ is all those numbers which completely divides the number $n$. For example, the factor of $12 = \{ 1,2,3,4,6,12\} $.
Greatest common divisor of two numbers $a$ and $b$ is the greatest number $c$ which divides both $a$ and $b$ completely. For example $GCD(6,16) = 2$. Co-prime numbers are those numbers such that their $GCD(a,b) = 1$.
Now, let’s find the factor of number $48$.
As we know that the factor of $48$ are
$48 = \{ 1,2,3,4,6,8,12,16,24,48\} $
It means that all numbers which are factors of $48$ have $48$ is their multiples.
So the list of pair of numbers which have $LCM$ equal to $48$ are
$(1,48),(2,48),(3,48),(4,48),(6,48),(8,48),(12,48),(16,48),(24,48),(48,48)$
We can also see that divisors $3$ and $16$ are co-prime numbers.
Co-prime numbers $LCM$ is equal to their product. So the numbers $3$and $16$also have $LCM$ equal to $48$. Now the multiples of $3$ and $16$ multiples in list of factors of $48$have their $LCM$equal to $48$.
So we add these pairs of numbers in a list of possible pairs of numbers which have $LCM$equal to$48$.
$(3,16),(6,16),(12,16),(24,16)$.
Hence the list of pairs of numbers whose $LCM$ equal
to$48$ be $(3,16),(6,16),(12,16),(24,16)$$,$$(1,48),(2,48),(3,48),(4,48),(6,48),(8,48),(12,48),(16,48),(24,4 8), (48,48)$

Note: While solving questions which are related to least common multiple, greatest common divisors. For solving these types of questions we need to know the definition of prime numbers, co-prime numbers, composite numbers, factors, multiples etc.