Question

LCM of $ 100 $ and $ 101 $ is?
(A) $ 10100 $
(B) $ 1001 $
(C) $ 10101 $
(D) None of these.

Answer 1

Hint: LCM or Least Common Multiple is defined as “the least number out of two or more numbers that are completely divisible by each of these numbers and do not leave any remainder.” We solve this question by “Prime Factorization Method”, in this method, to find the LCM of given numbers, we have to find their factors first and then multiply those factors to find the LCM of the numbers. For example, factors of 10 and 3 are given as,
 $
10 = 2 \times 5 \times 1\\
3 = 3 \times 1
 $
So, the LCM of 10 and 3
 $
{\text{ = }}2 \times 5 \times 3 \times 1\\
 = 30
 $

Complete step-by-step answer:
Given:
The given numbers are- 100 and 101
So, first calculating the factors of $ 100 $ , we have,
 $\Rightarrow 100 = 2 \times 2 \times 5 \times 5 \times 1 $
Similarly, calculating the factors of $ 101 $ we have,
 $\Rightarrow 101 = 101 \times 1 $
Since, $ 101 $ is a prime number, that is why it has only two factors $ 101 $ and $ 1 $ .
Now, multiplying the factors of $ 100{\text{ and 101}} $ to find the Least Common Factor or LCM. We get,
 $
\Rightarrow {\text{LCM = }}2 \times 2 \times 5 \times 5 \times 101 \times 1\\
{\text{LCM = }}10100
 $
Therefore, the LCM of $ 100 $ and $ 101 $ is $ 10100 $ and the correct option is (A) $ 10100 $

So, the correct answer is “Option A”.

Note: The alternative method of solving this question is by using the “GCF Method.” Where, GCF is the Greatest Common Factor of the two or more numbers, also known as the largest integer, which completely divides those numbers.
The formula used to calculate the LCM of two numbers by using this method is given by,
 $\Rightarrow {\text{LCM of two numbers = }}\dfrac{{{\text{Multiplication of two numbers}}}}{{{\text{GCF of two numbers}}}} $
So, GCF of $ 100 $ and $ 101 $ is calculated by using their factors, so-
The factors of $ 100 $ are, $ 100 = 2 \times 2 \times 5 \times 5 \times 1 $
And, the factors of $ 101 $ are, $ 101 = 101 \times 1 $
The only common factor between $ 100 $ and $ 101 $ is $ 1 $ .
So, the GCF of $ 100 $ and $ 101 $ is $ 1 $ .
Now, substituting the values into the formula we get,
LCM of 100 and 101
 $
\Rightarrow {\text{ = }}\dfrac{{{\text{100}} \times {\text{101}}}}{1}\\
 = 10100
 $
Therefore, using the GCF Method, the LCM of $ 100 $ and $ 101 $ is $ 10100 $ .