Question

The L.C.M of 27, 63 and 42 is
A) 926
B) 756
C) 998
D) 378

Answer 1

Hint: We will find the L.C.M of the given numbers using Table-Method. We will divide all the integers given by a prime number starting from least prime number 2 and keep this process going till we make all the integers 1. Then we will multiply all the prime numbers used for the division to get our L.C.M. L.C.M also known as the Least common multiple of two or more integers is the smallest number that can be divisible by all the integers given. It is a product of odd prime factors and common factors of the integers given and it is divisible by each integer.

Complete step-by-step answer:
First, we will list all the integers in a vertical table as below where all integers are denoted as \[x\].

\[x\]

\[27\]

\[63\]

\[42\]

Now we will divide all the numbers by a prime number starting from the least prime number 2 and form a new column for it.

\[x\]	\[2\]
\[27\]	\[27\]
\[63\]	\[63\]
\[42\]	\[21\]

Now as no integers in column 2nd are divided by 2 so we will take the next prime number that is 3 and form another column.

\[x\]	\[2\]	\[3\]
\[27\]	\[27\]	\[9\]
\[63\]	\[63\]	\[21\]
\[42\]	\[21\]	\[7\]

The integer in the third column is divisible by 3, so we will form a fourth column.

\[x\]	\[2\]	\[3\]	\[3\]
\[27\]	\[27\]	\[9\]	\[3\]
\[63\]	\[63\]	\[21\]	\[7\]
\[42\]	\[21\]	\[7\]	\[7\]

Again integers in the fourth column are again divisible by 3 we will form the fifth column.

\[x\]	\[2\]	\[3\]	\[3\]	\[3\]
\[27\]	\[27\]	\[9\]	\[3\]	\[1\]
\[63\]	\[63\]	\[21\]	\[7\]	\[7\]
\[42\]	\[21\]	\[7\]	\[7\]	\[7\]

Now as no integer in the fifth column is divisible by 3 we will take the next prime number that is 7 and form the sixth column.

\[x\]	\[2\]	\[3\]	\[3\]	\[3\]	\[7\]
\[27\]	\[27\]	\[9\]	\[3\]	\[1\]	\[1\]
\[63\]	\[63\]	\[21\]	\[7\]	\[7\]	\[1\]
\[42\]	\[21\]	\[7\]	\[7\]	\[7\]	\[1\]

As all the integers in the sixth column are 1, we will stop here.
Now we will multiple all the prime numbers in the first row to get our L.C.M.
L.C.M \[ = 2 \times 3 \times 3 \times 3 \times 7\]
\[ \Rightarrow \] L.C.M \[ = 378\]
Hence, option (D) is correct.

Note: Another method to find L.C.M is the Prime factorization method. In this method, we write each integer as a multiple of prime numbers and then multiply the highest power prime numbers together to get the LCM of the given numbers.
Therefore,
 \[\begin{array}{l}27 = {3^3}\\63 = {3^2} \times {7^1}\\42 = {2^1} \times {3^1} \times {7^1}\end{array}\]
Now we will multiple all the prime numbers having the highest power.
L.C.M \[ = {2^1} \times {3^3} \times {7^1}\]
\[ \Rightarrow \] L.C.M \[ = 2 \times 9 \times 7\]
\[ \Rightarrow \] L.C.M \[ = 378\]