Question

# Find the $L.C.M.$ of $30{\text{ and }}45$.A $15$B $30$ C $45$ D $90$

Hint:In the question where we have to find the L.C.M we will proceed by using the definition of L.C.M.If $a{\text{ and }}b$ are two non-zero integers the least common multiple of $a{\text{ and }}b$ is the smallest positive integer divisible by both $a{\text{ and }}b$.And then we will find the factors of the both numbers and find the smallest positive integer divisible by both the integers which will be the L.C.M of the given two integers.

In the above question we have to find the $L.C.M.$ of $30{\text{ and }}45$-
Now to find $L.C.M.$ of any two numbers we must know the definition of the $L.C.M.$
$L.C.M.$, least common multiple is defined as if $a{\text{ and }}b$ are two non-zero integers the least common multiple of $a{\text{ and }}b$ is the smallest positive integer divisible by both $a{\text{ and }}b$. The least common multiple is also known as smallest common multiple or lowest common multiple.
Now the least common multiple or the smallest common multiple of $a{\text{ and }}b$ is denoted by $l.c.m(a,b)$.
Now according to the question we have to find the $L.C.M.$ of $30{\text{ and }}45$, so now using definition of least common multiple, $30{\text{ and }}45$ are the non-zero positive integers so the $l.c.m(30,45)$ is the smallest positive integer divisible by both $30{\text{ and }}45$.
Now to find the smallest positive integer divisible by both $30{\text{ and }}45$, we have to find the factors the integers $30{\text{ and }}45$-
The factors of
$30 = 2 \times 3 \times 5 -------(1)$
$45 = 3 \times 3 \times 5 -------(2)$
Now from (1) and (2), we can see that
Common factor of $30,45$ $= 3 \times 5$
Non-common factor of $30,45$ $= 2 \times 3$
Now we know that the $L.C.M$ of $30{\text{ and }}45$ or smallest positive integer divisible by both $30{\text{ and }}45$ is equal to the product of Common factor of $30,45$ and Non-common factor of $30,45$.
So, mathematically we write that-
$l.c.m(30,45) = 3 \times 5 \times 2 \times 3$
$l.c.m(30,45) = 90 ------(3)$
Hence from (3), we get the least common multiple or $L.C.M$ of $30{\text{ and }}45$ is $90$.

So, the correct answer is “Option D”.

Note:In this type of question after writing the factors of the given two integers students generally make mistakes in finding the smallest positive integer divisible by the two given integers. So, to avoid this mistake one should firstly find the product of common factors and the non-common factors of the given integers, which will be equal to $l.c.m(30,45)$.
In this type of question, we can also be asked to find L.C.M of three integers, for that we will use the definition of L.C.M. and extend it to three integers.So, the least common multiple of three non-zero integers will be the smallest positive integer divisible by the three given integers.Therefore, we will find the factors of the three integers and find the L.C.M.
Also, in the case of three integers we can proceed like this also we will firstly find the L.C.M of any two integers and then finally the L.C.M all the three integers will be the L.C.M of the third integer and the obtained L.C.M of the two integers.