lcm and hcf of whole numbers

Rules of Divisibility

There are some simple rules of divisibility which enable us to find out whether a certain number is divisible by 2, 3, 4, 5, 6, 8, 9, 10 or 11

CLASS ACTIVITY

1. Using the rules of divisibility, find out which of the following numbers are divisible by
2. a) 2 b) 5 c) 4
3. i) 136 ii) 4 881 iii) 372 iv) 62, 784 v) 1010
4. Which of the following numbers are divisible by a) 3 and 9 b) 4 and 5?
5. a) 637 245 b) 134 721 c) 10140.

DEFINITIONS

EVEN NUMBERS: Even numbers are numbers that when divided by two has no remainder. All numbers that end in 0, 2, 4, 6, and 8 are even. Examples include: 34, 86, 26890, etc.

ODD NUMBERS: These set of numbers has a remainder of one when it is divided by 2. All numbers that end in 1, 3, 5, 7 and 9 are odd numbers. Examples are 81, 1247, 30096, etc.

COMPOSITE NUMBERS: These are numbers that are not prime numbers. They have factors other than 1 and the number itself. All even numbers except 2 are composite numbers.

FACTORS, MULTIPLES & THEIR RELATIONSHIP

FACTORS:  When two or more smaller numbers multiply to give a bigger number, these smaller numbers are called factors of the bigger number. In another sense we can say a factor is a number which can divide another number exactly without any remainder.

Examples:

• The factors of 24   are   1, 2, 3, 4 , 6 , 8 , 12 , and 24.
• The factors of  60   are   1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
• The factors of 50 are 1, 2, 5, 10, 25 and 50.

MULTIPLES: This is the product of numbers (factors) that gives other numbers.

Thus, 24 is:  a multiple of 1 twenty-four times.

a multiple of  2  twelve times.

a multiple of  3  eight times.

a multiple of  4 six times.

a multiple of  6  four times.

a multiple of  8  three times.

                                        a multiple of  12  two times.

a multiple of  24 (itself) once.

This shows the relationship between Factors and Multiples.

NOTE: The Teacher can make students do same analysis (orally) for 60 and 50 as has just been done for 24 above.

PRIME NUMBERS.

A prime number is a whole number that has only two factors which are 1 and the number itself. In other words, a whole number that has no other factor(s) except 1 and the number itself is referred to as a Prime Number. Number 1 or Integer 1 is not considered as a Prime Number.

Examples of Prime Numbers:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71,

 73, 79, 83, 89, 97 as those prime numbers between 1 and 100.

NOTE: Other higher ones should be listed also.

CLASS ACTIVITY

1:   List the factors of    (a).  48.    (b).  64.   (c)105  .

2:   48, 64, 108 are multiples of which numbers?

3:  Define a Prime Number; find the sum of all the prime numbers between 1 and 30.

DIFFERENCE BETWEEN FACTORS AND PRIME FACTORS

• The factors of 24   are   1, 2 , 3, 4 , 6 , 8 , 12 , and 24.   However, those factors that are Prime among all these are only 2 and 3. Hence, the Prime Factors of 24 are 2 and 3
• The factors of  60   are   1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.  However, those factors that are Prime among all these are only 2, 3 and 5. Hence, the Prime Factors of 60 are 2, 3 and 5
• The factors of 50 are 1, 2, 5, 10, 25 and 50. However, those factors which are Prime among all these are only 2 and 5.   Hence, the Prime Factors of 50 are 2 and 5

EXPRESSING NUMBERS AS PRODUCT OF PRIME FACTORS.

Examples:

1. Express 200 as product of prime factors in index form.

Solution:

200 =

2. Express 180 as product of prime factors in index form.

Solution:

3. Express 510 as product of prime factors in index form.

Solution:

.

CLASS ACTIVITY

1:  List the factors of 250 and the Prime factors of 250.

2:  List the factors and prime factors of 180.

3:  Express 252 as product of prime factors in index form.

4:   Express 440 as product of prime factors in index form.

5:   Express 15288 as product of prime factors in index form.

COMMON FACTORS AND HIGHEST COMMON FACTOR (H.C.F) OF TWO, THREE OR MORE NUMBERS.

        Worked Examples:  

1. Find the Common factors of 42 and 70.

Solution:

The factors of 42 are   1, 2, 3, 6, 7 , 14 , 21, 42.

The factors of 70 are   1, 2, 5, 7, 10, 14, 35, 70.

The factors that are common to both numbers or which are found in the two lists are: 1, 2, 7, 14.

The highest of the common factors here is 14. Hence, the Highest Common Factor (H.C.F ) of 42 and 70  = 14.

2. Find the Common Factors of 18, 27 and 36. What is their Highest Common Factor?

Solution:

The factors of 18 are 1, 2, 3, 6, 9, 18.

The factors of 27 are 1, 3, 7, and 27.

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Their Common Factors are:  1, 3.   Thus, their Highest Common Factor  ( H.C. F ) is 3.

Note: Teacher may improvise ALITERNATIVES (other methods) and demonstrate to learners in class.

 LEAST COMMON MULTIPLE (L.C.M) OF NUMBERS.

    Worked Examples:  

1. Find the Least Common Multiple (LCM) of 42 and 70.

  Solution:

Write 42 as product of prime numbers as follows:

42 =

Write 70 as product of prime numbers as follows:

70 =

Notice those numbers common to both set of prime numbers.  The common numbers are 2 and 7.

The Product of 2 and 7 gives 14.  Thus, in another way and by the way 14 is the Highest Common Factor

(H.C.F). But the L.C.M (Lowest Common Multiple) =

Therefore the L. C. M of 42 and 70 = 210.

2. Find the Least Common Multiple (LCM) of 18, 27 and 36.

Solution:

Write 18 =

Write 27

Write 36 =

NOTE: Teacher to assist Learners to read out those numbers which are to be selected and multiplied together from among the listed prime factors of the given numbers (as in above ) so as to arrive at the final correct L.C.M value.

In this example, the numbers to be picked for L.C.M are

Therefore the Least Common Multiple of 18, 27 and 36 =

NOTE: Teachers should ensure the difference between LCM and HCF is appreciated at the course of teaching these topics

CLASS ACTIVITY

1. Find the Common Factors of 60 and 84.  State the Highest Common Factor.
2. What is the Lowest Common Multiple of (L.C.M) of 60 and 84?
3. Find the L.C.M and H.C.F of 42, 90 and 105.

QUANTITATIVE APTITUDE REASONING ON LCM

Sample:

2³x2³x7 = 448

2×3²x5 = 90

2²x3² = 36

Example:

Find the missing number: ?? x 3 x 5 x 7 = 1680

Solution:

Let the number be

Expressing 16 as a multiple of 2 in index form yields 2 x 2 x 2 x 2 = 2⁴

Therefore 2⁴ x 3 x 5 x 7 = 1680

CLASS ACTIVITY

Do the following:

1. ?? x 3 x 5 = 60
2. 2 x ?? x 5 = 6480
3. 5² x 7 x ?? = 1925

QUANTITATIVE APTITUDE REASONING ON HCF

Sample:

(a). 28 = 2 x 2 x 7 = 2² x 7   (b). 36 = 2 x 2 x 3 x 3 = 2² x 3²   (c). 24 = 2 x 2 x 2 x 3 = 2³ x 3

Example:  Find the missing number in 64 = ⁶

Solution:

64

Multiply 2 by it-self in 6 times gives 64.

the missing number is 2 . This implies 64 = 2⁶

ACTIVITY

Now do the following by supplying the missing number in each case:

1. 84 = ?? x 2 x 3 x 7
2. ?? = 2 x 3 x 5
3. ?? = 3² x 5²

PRACTICE EXERCISE

1. Given the numbers 3510, 7460, 4815, and 5645, state which of the numbers are:
2. a) Divisible by 3 b) divisible by 5 c) divisible by 15.
3. List the factors of: a) 45 b) 60 c) 120
4. Find the LCM and HCF of a) 20, 30 and 60; b) 32, 48 and 72
5. Express 72 as a product of its prime factors in index form
6. List the first five multiples of 7.

ASSIGNMENT

1. Which of the following numbers are divisible by 6?
2. a) 2352 b) 8134 c) 7812
3. List the common factors 45 and 60
4. Find the positive difference between the HCF of 24 and 36 and the LCM of 15 and 20.
5. Express 120 as products of its prime factors in index form.
6. Find the prime factors of a) 24 b) 60 c) 35.