Question

Find the L.C.M. of 24, 60, and 150 by fundamental theorem of arithmetic.

Answer 1

Hint:
Here, we will write the given numbers as a product of their prime factors. Then, we will calculate the L.C.M. of the three numbers. L.C.M. or the lowest common multiple is the product of the prime factors with the greatest powers.

Complete step by step solution:
The fundamental theorem of arithmetic states that every composite number can be written as a product of its prime factors in a unique way.
A prime factor is a factor of a number which is divisible by 1 and by itself.
First, we will write the given numbers as a product of their prime factors.
We know that 24 is the product of 8 and 3.
Therefore, we can write 24 as
\[ \Rightarrow 24 = 8 \times 3\]
8 is the cube of the prime number 2. Thus, we get
\[ \Rightarrow 24 = {2^3} \times 3\]
Now, we know that 60 is the product of 3, 4, and 5.
Therefore, we can write 60 as
\[ \Rightarrow 60 = 3 \times 4 \times 5\]
4 is the square of the prime number 2. Thus, we get
\[ \Rightarrow 60 = 3 \times {2^2} \times 5\]
Next, we know that 150 is the product of 2, 3, and 25.
Therefore, we can write 150 as
\[ \Rightarrow 150 = 2 \times 3 \times 25\]
25 is the square of the prime number 5. Thus, we get
\[ \Rightarrow 150 = 2 \times 3 \times {5^2}\]
Therefore, we have
\[24 = {2^3} \times 3\]
\[60 = 3 \times {2^2} \times 5\]
\[150 = 2 \times 3 \times {5^2}\]
Now, in the product of primes, we can observe that the greatest power of 2 is 3, greatest power of 3 is 1, and the greatest power of 5 is 2.
Thus, the prime factors with the greatest powers are \[{2^3}\], 3, and \[{5^2}\].
The lowest common multiple of the numbers 24, 60, 150 is the product of the prime factors with the greatest powers.
Therefore, we get
\[L.C.M. = {2^3} \times 3 \times {5^2}\]
Simplifying the expression, we get
\[\begin{array}{l} \Rightarrow L.C.M. = 8 \times 3 \times 25\\ \Rightarrow L.C.M. = 600\end{array}\]

\[\therefore\] The L.C.M. of 24, 60, and 150 is 600.

Note:
We have to remember that all the prime factors with the greatest powers are selected, irrespective of whether that power appears in the prime factorization of all the three numbers 24, 60, and 150. For example, \[{2^3}\] does not appear in the prime factorization of 60 and 150. But it should be included while calculating L.C.M. because it has the highest power. Another common mistake is to use the common factors with the lowest powers to calculate the L.C.M. That is incorrect because it will give you the H.C.F. and not the L.C.M. of the numbers.