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General Mental Ability Notes-LCM, HCF-For W.B.C.S Exam.
সাধারণ মানসিক ক্ষমতা নোটস – এলসিএম, এইচসিএফ-WBCS পরীক্ষার জন্য ।
What is Prime number?
- Consider this number : 12. This number can be found in many multiplication tables for example.Continue Reading General Mental Ability Notes-LCM, HCF-For W.B.C.S Exam.
- 1 x 12=12.
- 2 x 6 =12
- 3 x 4=12
- That means, 12 has many factors (1,2,3,4,6,12). Such number is called a composite number.
- On the other hand, consider this number: 29. You cannot find it in any table except 29 x 1 =29. Such number is called a prime number.
- Let’s make a shortlist from exam point of view
| Prime | Non-prime (composite) |
| 2,3,5,7,11,13,17,19,23,29 | 4,6,8,9,10,12,14,15…. |
Now hold this prime number thought in your mind for a while.
What is LCM?
First, let’s create multiplication tables of 4 and 6.
| 4’s table | multiple | 6’s table | multiple |
| 4 x 1 = | 4 | 6 x 1 = | 6 |
| 4 x 2 = | 8 | 6 x 2 = | 12 |
| 4 x 3 = | 12 | 6 x 3 = | 18 |
| 4 x 4 = | 16 | 6 x 4 = | 24 |
| 4 x 5 = | 20 | 6 x 5 = | 30 |
| 4 x 6 = | 24 | 6 x 6 = | 36 |
| 4 x 7 = | 28 | 6 x 7 = | 42 |
| 4 x 8 = | 32 | 6 x 8 = | 48 |
| 4 x 9 = | 36 | 6 x 9 = | 54 |
- Do you see any common numbers in the multiples of 4 and 6?
- Yes I see 12, 24 and 36 are common in both tables. Let’s isolate them.
| 4 x 3 = | 12 | 6 x 2 = | 12 |
| 4 x 6 = | 24 | 6 x 4 = | 24 |
| 4 x 9 = | 36 | 6 x 6 = | 36 |
- 12, 24 and 36 are common multiples of 4 and 6. But what is the smallest of these multiples?
- Ans 12 is smallest.
How to find LCM using Prime-Factorization?
Suppose in the exam, we need to find LCM of 4 and 6.
Make a table like this
| Number | Factors |
| 4 | |
| 6 |
Now you need to find the prime factors of 4 and 6.
| Number | Factors |
| 4 | 2 x 2 |
| 6 | 2 x 3 |
Express it in terms of “powers”. For example 2 x 2 =22
| Number | factors |
| 4 | 22 |
| 6 | 2 x 3 |
Now make the third row called “LCM”.
| Number | factors |
| 4 | 22 |
| 6 | 2 x 3 |
| LCM |
Now write all prime numbers in this “LCM row”
| Number | factors |
| 4 | 22 |
| 6 | 2 x 3 |
| LCM | 2, 3 |
Write maximum power of each prime number
| Number | factors |
| 4 | 22 |
| 6 | 2 x 3 |
| LCM | 22, 3 |
As you can see, maximum power of 2 was 22 (in 4’s row).
Now multiple the numbers given in LCM row
| Number | factors |
| 4 | 22 |
| 6 | 2 x 3 |
| LCM | 22 x 3 =12 |
That’s our answer. LCM (4,6)=12.
LCM of two numbers (56, 96)
| Numbers | Factors |
| 56 | |
| 96 |
| Number | Factors |
| 56 | 8 x 7 |
| 96 | 12 x 8 |
- but we need factors in “prime number” format. 12 and 8 are not prime numbers. So let’s Simplify further.
- 56 = 8 x 7 = 23 x 7 (; because 8 = 4 x 2 = 2 x 2 x 2)
- 96 = 12 x 8 = (4×3)x(4×2)=( 22x3) (23)=25x3
| Number | Factors |
| 56 | 23 x 7 |
| 96 | 25x3 |
Now let’s make the LCM row. Write all prime numbers (2,3 and7) in ascending order.
| Number | Factors |
| 56 | 23 x 7 |
| 96 | 25x3 |
| LCM | 2 3 7 |
Now write maximum powers of each prime number.
| Number | Factors |
| 56 | 23 x 7 |
| 96 | 25x3 |
| LCM | 25 3 7 |
Multiply these numbers
| Number | Factors |
| 56 | 23 x 7 |
| 96 | 25x3 |
| LCM | 25 x3x7=32×21=672 |
So LCM (56,96)=672
LCM of three numbers: (12,15,20)
Approach is same. Make prime factors
| Number | Prime factors |
| 12 | 22 x 3 |
| 15 | 3 x 5 |
| 20 | 22 x 5 |
Make a new row, write all prime factors in ascending order.
| Number | Prime factors |
| 12 | 22 x 3 |
| 15 | 3 x 5 |
| 20 | 22 x 5 |
| LCM | 2,3,5 |
In the last row, Write the maximum power of those prime numbers.
| Number | Prime factors |
| 12 | 22 x 3 |
| 15 | 3 x 5 |
| 20 | 22 x 5 |
| LCM | 22, 3, 5 |
Now multiple the numbers in last row
| Number | Prime factors |
| 12 | 22 x 3 |
| 15 | 3 x 5 |
| 20 | 22 x 5 |
| LCM | 22x3x5=60 |
Therefore LCM (12,15,20)=60.
You can also look at it in following way-
- 12 x 5 = 60
- 15 x 4 = 60
- 20 x 3 = 60.
So 60 is the least common multiple.
LCM of prime numbers
Find LCM of 7,11,13
| Numbers | Factors |
| 7 | 7 x 1 |
| 11 | 11 x 1 |
| 13 | 13 x 1 |
| LCM | 1x 7 x 11 x 13 =1001 |
So 1001 is the answer.
LCM of co-prime numbers
- Co prime numbers are those numbers that donot have any common factors. For example, 14 and 15.
- Individually none of them is prime number because 14=2 x 7 and 15 = 3 x 5.
- But they (14 and 15) donot have any common factors. So they’re called co-prime numbers (when they’re given together).
- Any two consecutive numbers are co-prime numbers. (e.g. 11,12 or 1548,1549).
- In case of co-prime numbers, just multiply them and you will get LCM. There is no need to find factors. example
| 6 | 2 x 3 |
| 7 | 7 |
| LCM | 2 x 3 x 7 = (6)x7 =42 |
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