Least common multiplier, Greatest common Divider

Least common multiplier [LCM]

           The Least common multiplier [LCM] calculated for two or more numbers for small applications like food bags in particular specified area, arrangement of the students  in the playground area in the form of columns and rows etc.

            The LCM of any Two numbers can be found by dividing the numbers with least prime numbers and taking one time the common prime factors and product all prime factors of the numbers. in this case divisibility Technics are useful.

Eg: LCM of 3 and 4

For 3 and four there are no common prime factors

Hence the LCM=product of the numbers=3x4=12.

Conclusion: If there is no common factor ,the LCM will be product of the numbers

E.g 2] Find the LCM of 2, 4

2	2,        4
2	1,        2
	1,        1

Take 1^st prime number 2

LCM=2x2=4

The least common multiplier=4

E.g 2] Find the LCM of  3, 9, 18

2	3,        9,    18
3	3,        9,     9
3	1,        3.      3.
	1,        1,      1

Take 1^st prime number 2

Not divisible 2 so take 3

LCM of the numbers=2x3x3=18

Conclusion: If all numbers given are factor of the one big number among the numbers, the bigger number will be LCM.

E.g3: Find the LCM of 6,9 and 15

2	6,       9,       15
3	3,       9,        15
3	1,       3,         5
5	1,       1,          5
	1,       1,          1

Take the 1^st prime number

2 is not disable hence take next prime number 3

Do the same process until we get 1 in last row.

LCM=2x3x3x5=90

The least common multiplier=90

I.e the numbers 6, 9, 15 are divides the LCM 90 commonly with least quotients.

Check it   by 6       90/6=15

                By 9       90/9=10

                By 15     90/15=6

  Conclusion : The quotients are not having common divider.

• If there is no common factor among the given numbers, the LCM will be the product of the numbers.
• If one big number is divisible by remaining numbers given, the bigger number will be LCM.

Will be continue with some practical problems and solutions

Greatest Common Divider[GCD]

Greatest Common Divider [GCD] is calculated some practical solution like easy measurement of largest quantities with the greatest common measurement unit.

Eg. 1] GCD of two numbers    15, 6

This can be calculated by dividing the bigger number with smaller number and further dividing the smaller number with the remainder by dividing bigger number dropdown continuously.

 Take the 6 as divider                   6] 15 [2

                                                           12      

Drop down 6                                       3] 6 [2

 Divide with previous reminder 3            6

                                                                0

The reminder 0, hence the GCD=3

Calculate the LCM of 6,15 we get 30

The product of the numbers=6x15=90

The product of the LCM and GCD=30x3=90

Conclusion: The product of the given numbers =The product of the LCM and GCD of the numbers.

Examples
1] How many minimum number of the students can arrange in the form of 26 rows or 91 rows?

Sol: the number of rows of soldiers= 26 or 91

The minimum number of students can be arranged in 26 or 91 rows=LCM of 26, 91
                                                                           

2	26,    91
7	13,    91
13	13,   13
	1,       1

The LCM of the 26 and 91=2x7x13=182

The minimum students required for arrange as 26 or 91 rows in the play ground=182.

Example2] Find the least number that can be divisible by 12, 18, 28

            The numbers are 12,18, and 28

the least number that can be divisible by 12, 18, 28

2	12,     18,       28
2	6,       9,        14
3	3,       9,         7
3	1,       3,          7
7	1,       1,          7
	1,       1,           1

=The LCM of the numbers

The LCM of the numbers=2x2x3x3x7

                                        =252

The least number that can be exactly divisible by

12, 18 and 28=252

Example 3] Find the minimum number if the number divided by 7, 11 and 13, the reminder will be 5.

The numbers are 7, 11, and 13

The numbers are prime numbers i.e no common factor except 1

So LCM=Product of the numbers

          = 7x11x13=1001

For obtaining reminder as 5, add 5 to LCM.

Therefore the number=1001+5=1006.

Fun: Product of any three digit number and 7x11x13 gives six digit number that repeats the same number in the next three places. E.g 547x7x11x13=547547

Example 4 ] Find the minimum number if the number divided by 7, 11 and 13, the reminders will be 4, 8 and 10 respectively.

The numbers are 7, 11, and 13

The numbers are prime numbers i.e no common factor except 1

So  LCM of 7, 11, 13 [prime numbers]=Product of the numbers

          = 7x11x13=1001
The common difference=7-4=11-8=13-10=3

Therefore the number=1001-3=998

Check: 7]998[142

                 7

                 29

                 28

                  018

                    14

Reminder : 04

          

            11]998[90

                 99

                 08

                   0

Reminder:08

           13]998[76

                91

                 88

                 78

Reminder:10

The reminders are 4,8 and 10,so the solution justified.