Lowest Common Multiple 
& Highest Common Factor

  • prime number is a whole number that has exactly 2 different factors, 1 and itself.
  • composite number is a whole number that has more than 2 different factors.
  • The process of expressing 18 as a product of its prime factors is called the prime factorisation of 18.
  • The Lowest Common Multiple (LCM) of two or more numbers is the smallest multiple that is common to all the numbers.




Multiples are numbers that result from multiplying by an integer.
Ex. Multiples of 4 are 4,8,12,16,20.... the 4 times table.

Factors are numbers that can be multiplied to get another number. 

Ex. Factors of 6 are share 1,2,3, and 6.

Lowest Common Multiple:
LCM of 6 and 14. 
6: 6, 12, 18, 24, 30, 42
14: 14, 28, 42

LCM = 42
Highest Common Factor:
To find the HCF, list the factors of both numbers, identify the common factors in both lists, and then find the highest common number.
Ex. Find the HCF of 12 and 28:
12: 1,2,3,4,5,6,12
28: 1,2,4,7,14,28
HCF = 4

Q. Word Problem:

A park has two different light displays, one display lights up every 12 minutes, and the other lights up every 15 minutes. If both displays light up at 6:00 pm, when will they next light up at the same time?

Solution:
Multiples of 12: 12, 24, 36, 48,60
Multiples of 15: 15, 30, 45, 60

LCM= 7 pm















Digits​: 

The figures which make up a number are called digits.

Q.1. What is factor?

          Ans: The numbers by which a larger number can be divided are the factors of that number.

          Q.2. What is G.C.D?

            Ans: The greatest divisor that lies among the common factors of two numbers is the G.C.D.

           Q. 3. What is multiple? 

            Ans: When a number is divisible by another number, the first number is called the multiple of the other number.



 For example, in the number 627, 6 is called the hundred digit, 2 is called the ten-digit, and 7 the unit digit. 

A multiple of a number is exactly divisible by that number. 

For example, 24 is a multiple of 4. If a number is multiple of a second number, the second number is said to be a factor of the first. 

A prime number is a number which has no factors other than itself and unity. 

The first five prime numbers are 2, 3, 5, 7, 11. 

A prime factor is a factor which is a prime number.

 The prime factors of 24 are 2, 2, 2, and 3. 

Expressed in its prime factors , 

     24 = 2 x 2 x 2 x 3 

Ex. 7.4



 

1. List 3 multiples for the following numbers, ordering them from the smallest.

a) 4

Solution:   4 x 1 = 4

                   4 x 2 = 8

                  4 x 3 =  12

          2. List 3 common multiples for the following pairs of numbers, ordering them from the smallest. also write the least common multiples (LCM).

a) 3, 4

Solution:  Multiples of 3 :  3, 4, 6, 9, 12, 15, 18, 24, 27, 30, 33, 36,............

                  Multiples of 4 : 4, 8, 12, 16, 20, 24, 28, 32, 36,...........

            3 common multiples of 3 and 4 are 12, 24, 36

           Ordering from smallest to highest   12, 24, 36

            The smallest common multiple is 12

             The LCM of 3 and 4  = 12 Ans.



4. List all the common factors of the following numbers. also write the highest common factors (HCF) for each pair.

a) 9, 15

Solution: 9 = 1 x 9 = 3 x 3

             The factors of 9 :  1, 3, 9

           15 = 1 x 15

                 = 3 x 5

The factors of 15 :   1 ,3, 5, 15

The common factors of 9 and 15 are 1 and 3.

The highest common factors (HCF) for 9 and 15 is 3. Ans. 


5. Find the least common multiple (LCM) and highest common factors (HCF) for each .

a) 8, 12, 24

Solution:      8 = 4 x 2 

                        = 2 x 2 x 2

           12 = 2 x 6

               = 2 x 2 x 3

     24 = 2 x 12

         = 2 x 4 x 3

        = 2 x 2 x 2 x 3

    LCM   = 2 x 2 x 2 x 3 = 24

Again, 8 = 1 x 8

               = 2 x 4

and 12 = 1 x 12

           = 2 x 6

           = 3 x 4 

and 24 = 1 x 24

            = 2 x 12

           = 3 x 8

         = 4 x 6

HCF  = 4

Highest​ Common​ Factor​:

The Highest Common Factor (H.C.F) of two or more numbers is the greatest number which is a factor of each of them. 

Ex. Find the H.C.F and L.C.M. of 63, 441, 3969. 


Solution: 63 = 3 x 21 

   = 3 x 3 x 7 441 

 = 3 x 147

 = 3 x 3 x 49

 = 3 x 3 x 7 x 7 3969

 = 3 x 1323 

 = 3 x 3 x 441 

 = 3 x 3 x 3 x 3 x 7 x 7 

H.C.F. = 3 x 3 x 7 = 63 

L.C.M. = 3 x 3 x 3 x 3 x 7 x 7 

          = 3969 

       Lowest​ Common​ Multiple​:

The Lowest Common Multiple (L.C.M.) of two or more numbers is the least number which is a multiple of each of them.

Real-life Problem involving LCM:


Q. Find the lowest common multiple of 30 and 36.
Solution: 
              common prime factors
                        30  = 2   x 3               x  5
                         36 =   2  x 2 x 3  x 3
                     ________________________
LCM of 30 and 36 = 2 x 2 x 3    x 3     x 5
                              = 180






Q.1. Three bells toll at regular intervals of 15 minutes, 16 minutes and 36 minutes respectively. Given that they toll together at 2.00 p.m. at what time will they next toll together?

                                              15 = 5 x 3
                                              16 =  4 x 2 x 2
                                               36 = 3 x 3 x 2x 2 

                           LCM of 15, 16 , 36  = 2 x 2 x 3 x 5 x 4 x 3
                                                            = 720 minutes
                                   
                                                        720/60
                                                       = 12  hours

            Q.2.  Farhan has three pieces of rope with lengths of 140 cm, 168 cm and 210 cm. He wishes to cut all the three pieces of ropes into smaller pieces of equal length such that there is no leftover rope.
             i) What is the greatest possible length of each of the smaller pieces of rope?
            ii) How many smaller pieces of rope can he get altogether?







Q.13. The lights on three lightships flash at regular intervals. The first light flashes every 18 seconds, the second every 30 seconds and the third every 40 seconds. The three lights flash together at 10.00 p.m. At what time will they next flash together?

                               18 = 2 x 3 x 3 
                               30 = 2 x 3 x 5
                              40 = 2 x 2 x 5
                     LCM  of 18, 30 and 40  = 2 x 2 x 2 x 3 x 3 x 5
                                            = 360
                     360 seconds = 6 minutes
         The three lights will next flash together at 10.06 p.m.

Q.7. Huixian needs to pack 171 pens, 63 pencils and 27 erasers into identical gift bags so that each item is equally distributed among the gift bags. Find 
i) the largest number of gift bags that can be packed
ii) the number of each item in a gift bag.

Q. 6. Shirley needs to pack 108 stalks of roses, 81 stalks of lilies and 54 stalks of orchids into identical baskets so that each type of flowers is equally distributed among the baskets. Find
i) the largest number of baskets that can be packed

ii) the number of each type of flowers in a basket 

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