determine the LCM of the following numbers by division method 6 8 45
Answers
L.C.M. by Long Division Method:
A least common multiple of two numbers is the smallest positive number that is a multiple of both.
Multiple of 3 — 3, 6, 9, 12, 15, 18,…………..
Multiple of 4 — 4, 8, 12, 16, 20, 24,………….
So the LCM of 3 and 4 is 12, which is the lowest common multiple of 3 and 4.
An example of LCM
The LCM of 10, 20,25 is 100. It means that 100 is the lowest common multiple of these three numbers, but there is a question in our mind that can the LCM be (-100)? Since (-100) is lower than 100 and divisible by each of 10, 20, 25, or can it be zero or what will be the LCM of (-10) and 20? Will it be (-20) or (-200)?
For all these questions, there is only one answer that the LCM is only defined for positive numbers and LCM is not defined for 0.
PROCESS OF FINDING LCM
We will do prime factorization in first step of all the numbers.
Then we calculate the number of times each prime occurs in prime factorization and write each number as power of primes.
Then in last step we write all the primes involved and raise each of the primes to highest power present.
Example based on above
Example 1:LCM of 10, 20, 25?
Step 1: 10= 2 × 5
20 = 2 × 2 × 5
25= 5 × 5
Step 2: 10= 21 × 51
20 = 22 × 51
25= 52
Step 3:Primes involved are 2 and 5
Now we raise each of the primes to highest power present i.e.22 × 52 =100. So 100 is required LCM.
Example 2: What is the LCM of 35, 45, 55?
Step-1:
35 = 5 × 7
45 = 3 × 3 × 5
55 = 11 × 5
Step-2:
35 = 51 × 71
45 = 32 × 51
55 = 111 × 51
Step-3: Primes involved are 3, 5, 7 and 11
Now we raise each of the primes to the highest power present i.e. 32 × 51 × 71 × 111
LCM of the given numbers = 3465
Answer:
Hint: We need to find the least common multiple of 6, 8 and 45. We will take the simultaneous factorisation of those three numbers to find the LCM.
Complete step-by-step answer:
We need to find the LCM of 6, 8 and 45. LCM stands for least common multiple.
We first find the multiples of 6, 8 and 45.
We use the simultaneous factorisation to find the greatest common factor of 6,8 and 45.
We have to divide both of them with possible primes which can divide both of them.
2∣∣6,8,45−−−−−−3∣∣3,4,45−−−−−−2∣∣1,4,15−−−−−−2∣∣1,2,15−−−−−−3∣∣1,1,15−−−−−−5∣∣1,1,5−−−−−1∣∣1,1,1−−−−−
The LCM is 2×3×2×2×3×5=360.
Therefore, the least common multiple of 6, 8 and 45 is 360.
So, the correct answer is “360”.
Note: We need to remember that the LCM has to be only one number. It is the least common multiple of all the given numbers. If the given numbers are prime numbers, then the LCM of those numbers will always be the multiple of those numbers. These rules follow for both integers and fractions.