What is the Least Common Multiple (L.C.M) of two or more numbers?

The least common multiple which is shortened as (L.C.M) of two or more numbers is the smallest counting number that is a multiple of each of the giving numbers.

For example, let us see how to find the L.C.M of 3, 4 and 6.

Set of multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24 ...}

Set of multiples of 4 = {4, 8, 12, 16, 20, 24 ...}

Set of multiples of 6 = {6, 12, 18, 24, 30, 36 ...}

Go ahead find the Set of Common Multiples = {12, 24, 36 ...}, therefore the L.C.M of 3, 4 and 6 is 12. The L.C.M can also be found using Prime Factorization.

For instance if we want to find the L.C.M of 12 and 16,

12        =          2 x 2 x 3          =          2² x 3

16        =          2 x 2 x 2 x 2    =          2⁴

            To find the L.C.M, we need the prime factors whose product is a multiple of both 12 and 16. In this case, students will need each of the different from both factorization and also each to the highest power in which it occurs in either factorization. Hence the L.C.M of 12 and 16 is 2⁴ x 3 = 16 x 3 = 48.

            If a pair of numbers is relatively prime their H.C.F would be 1 but their L.C.M will be given by the product of the numbers. For example, the numbers 5 and 7 are relatively prime hence, the H.C.F of 5 and 7 is 1 and the L.C.M of 5 and 7 is given by 5 x 7 = 35.

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What is the Highest Common Factor (H C F) of two or more numbers?

The Highest Common Factor which is shortened as H.C.F of two numbers is the highest number in the set of Common Factors.

Helping students find the H.C.F of numbers, say 24, 36 and 48; three methods could be employed and they are as follows:

i) Using set of factors        ii) Using prime factorization        iii) Using common division.

Method 1: Using set of factors

NB: Having problems on Factors of Numbers, Visit: what is a Factor of a number? on my blog for clarification. Now,

Set of factors of 24 = {1, 2, 3, 4, 6, 8, 12, 24}

Set of factors of 36 = {1, 2, 3, 4, 6, 12, 18, 36}

Set of factors of 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}

Go ahead find the Set of Common Factors = {1, 2, 3, 4, 6, 12}

Therefore the highest common factor = 12.    Hence H.C.F of 24, 36, and 48 is 12.

Method 2: Using prime factorization

Hint: Using the knowledge of indices to represent prime factorization.

24        =          2 x 2 x 2 x 3    =          2³ x 3

36        =          2 x 2 x 3 x 3    =          2² x 3²

48        =          2 x 2 x 2 x 2 x 3 =       2⁴ x 3

Now it can be seen that 3 is common to all factorization, 2² is also common to all factorization. Hence the H.C.F = 2² X 3 = 12

Method 3: Using common division

NB: No number can go into 2, 3, and 4 at the same time, hence H.C.F is 12.


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Numbers Divisibility Rules

Divisibility rules help us to easily test if one number can be evenly divided by another. But before we continue, let us go ahead explain what is meant by “Divisible by”.

 Divisible By

"Divisible By" means "when you divide one number by another the result is a whole number"

Examples:

14 is divisible by 7, because 14÷7 = 2 exactly

But 15 is not divisible by 7, because 15÷7 = 2 ¹/₇ (i.e., the result is not a whole number)

Note that “Divisible by" and "can be evenly divided by" mean the same thing.

Study the following carefully:

Considering the multiples of 2 = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 ...} we see that the last digit is either zero or divisible by 2.

Considering the multiples of 3 = {3, 6, 9, 12, 15, 18, 21, 24, 27 ...} when we add the digits of any of the multiples, we get a number which is divided by 3, e.g. for 15 we have 1 + 5 = 6, 21 we have 2 + 1 = 3, and for 27 we have 2 + 7 = 9.

Considering the multiples 0f 11 = {11, 22, 33, 44, 55, 66, 77, 88, 99, 121, 132 ...} here, find the difference between the sum of its odd and even digits e.g. sum of odd digits 1 + 1 = 2, sum of even digits = 2 difference = 2 – 2 = 0.  For 132: sum of odd digits = 1 + 2 = 3, even digit = 3 difference = 3 – 3 = 0.

Considering the multiples of 4, 5, 6, 7, 8, 9 and 10, can a general statement involving patterns about these multiples be made?

Yeah, we may discover the following pattern about the multiples of the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11 which we called divisibility tests.

The patterns are tabulated as follows:

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What is a Prime Number?

Consider the table below,

It is clear from the table above that the numbers 2, 3, 5, 7 ... have two numbers as their factors. Such numbers are called Prime Numbers. Therefore a Prime Number is counting number having exactly two factors, one and itself. Note that numbers that have more than two factors are called composite numbers. E.g. 4, 6, 8 and so on.

NB: this it can be helpful.

Prime Factorization:

When we talk of Prime Factorization, it is the way of expressing a number as a product of prime number. “Factor tree” could be used to find the prime factorization of numbers.

This is often used and most people enjoy it! For example to find the prime factorization of 42 and 48 using the factor tree.

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Multiples of Numbers

It can be seen that 2 is a factor of 6. Supposing that 2 is a factor of 6 then we say that 6 is a multiple of 2. This shows that if a number ‘a’ divides a number ‘b’ exactly then ‘a’ is a factor of ‘b’ and ‘b’, a multiple of ‘a’. For example, 4 is a factor of 20 that means, 20 is a multiple of 4. The table below shows more examples on set of multiples of numbers.

Note that the set of multiples of numbers are infinite set.



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Even and Odd Number: what is the difference?

Consider the table below:

Note:  From the table, some numbers have 2 as a factor and others do not have 2 as a factor.

Numbers like {2, 4, 6, 8, 10, 12 ... } which have 2 as a factor are called Even Numbers. Also numbers like {1, 3, 5, 7, 9, 11 ...} which do not have 2 as a factor are called Odd Numbers.

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What is a Factor of a number?

Suppose one number divides a second number exactly then the first number is a factor of the second. For example 3 divides 12 into 4. We can say that 3 and 4 are factors of 12.

To know that 12 has order factors, let us look at the table in fig 1

This shows all number that make the statement a x b= 12 true to be the factor of 12.

Therefore, the set of factors of 12 is {1, 2, 3, 4, 6, and 12}. The set consist of 6 numbers.

Fig 2 shows the first ten counting numbers and their set of factors. Students have to study them cautiously.

Natural numbers and Whole Numbers: What is difference between them?

Most students get confuse with Natural numbers and Whole numbers. I hope this tutorial will get rid of this confusion.

Now, the set of Natural numbers (N) is given by: N= {1, 2, 3, 4 ...} and the set of Whole numbers (W) is given by: W = {0, 1, 2, 3, 4 ...}. Thus the intersection of N and W = N and the union of N and W = W.

The smallest natural number is 1 and there is no largest natural number.

The smallest whole number is 0 and there is no largest whole number.



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Number We Use Today

One may ask; what is a number? A number expresses an idea of quantity. It cannot be seen as a concrete object as a stone or paper, but it can be made real by representing it by a word or a symbol. These symbols which are used to represent numbers are called numerals. We have numeration system from Africa (Egypt), Europe (Rome), and Asia (Babylon), which was used some time ago. Now the numeration system use today is from the Hindu-Arabic system.

Hindu-Arabic Numeral

The Hindu-Arabic Numeral is numeral we use today. It was originated in India and was later adopted by Arab traders. The Hindu-Arabic numeral used only ten symbols called digits, to write all numbers. The symbols, as we know them today are as follows:

(i)                 The ideal of place value                (ii) Symbol for zero.

With the ideal of place value and the symbol for zero, we can write many numbers from the digits. For example 178, 136, 675, 879, 354, are all numbers which use the same digits but different place in the numeral allows different numbers to be named. The symbol ‘0’ in the digits is used for two purposes:

(i)                 As the number for no quantity                  (ii) to serve as a place value holder.

Consider the numbers 7, 700, 7000, 70000, 700000 and so on. All the numbers are written with 7, the zero brings the difference in the numbers.

The Hindus group units together to make tens, group tens together to make hundreds and so on. A number 753 represents seven hundred plus five tens plus three ones.

The Hindu-Arabic Numeral is sometimes called the base ten numeral.

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Know the Basic Number Concept

In this tutorial, we may look at the concept of numbers from three different views as:

Ø  Cardinal number concept

Ø  Ordinal number concept

Ø  Ratio number concept

The Cardinal Number Concept:

This number concept expresses quantity. When we count the number of elements in the set, we are talking about the cardinal number concept. E.g. 2, 4, 8 and so on.

The Ordinal Number Concept:

This number concept expresses position. It involves ranking of numbers. When we finding the position of an event e.g. 1^st person in class test, 3th person in a race, 5^th January etc. The numbers 1^st, 3^rd, 5^th, 7^th, etc express the ordinal of number.

The Ratio Number Concept:

This number concept is comparing number with standard measure. It involves measurement. The number is attached to measurement in the ratio number concept. If a book is 4cm long then the book is 4cm compared to a standard centimetre. The numbers 2meters, 4cm, 5kg, 9litres etc express ratio concept of numbers.

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