Factors and Multiples
Category : 5th Class
This lesson will help you to:
Real - Life Example
QUICK CONCEPT REVIEW
It was picture day in Ria's school. Her teacher made all the students stand in a single line. But all of them couldnot come in the frame.
This way also all the students were not fitting in the frame.
Then she made 4 lines of 5 each. Now all the students could fit in the frame.
So here we saw three different ways to make 20 students stand in lines.
The first way is \[1\times 20\]
The second way is \[2\times 10\]
& the third way is \[5\times 4\]
Therefore, we can say that 1,20,2, 5 & 4 are the factors of 20.
Definition of factors: The factors of a number are thosewhich divide the number without leaving any remainder.
Thus, factors of a number divide the number completely,
Note: A number can have many factors.
Example 1: A factor tree of 8 is given in figure A. Here 8 has been broken into 2 factors 4 A 2. But 4 is not a prime number. 4 is again broken into 2 factors 2 & 2 as shown in
figure - A. Therefore, the factors of 8 are 2, 2 A 2.
Example 2: We can make factor trees of a same number 60 in different ways as shown in figures - B, C and D:
Fig.-B: This is a factor tree of 60. Here 60 has been broken into two factors 6 & 10. But 6 & 10 are not prime numbers. 6 & 10 are again broken into two factors each. 6 is broken in 2 & 3 and 10 is broken into 2 & 5.
Fig.-C: This is also the factor tree of 60. Here 60 has been broken into 3 & 20. 20 is further broken into 4 & 5. Again 1 more branch is added to thisfactor tree by breaking 4 into 2 & 2.
Fig. D: This is another factor tree for 60. Here 60 has been broken into 2 & 30. 2 is a prime number hence we will leave it as it is and further break30 into two factors 2 & 15.15 is again broken intotwo factors 3 & 5. Now 3 & 5 are prime numbers,therefore, the tree is complete now. That is, we cannot add any more branch to it.
The same factors of two or more than two differentnumbers are called common factors.
Let us find out the factors of 15& 21.
Factors of 15 are 1, 3, 5 & 15.
Factors of 21 are 1, 3, 7 & 21.
Therefore, Common factors of 15 & 21 are 1 and 3.
HIGHEST COMMON FACTOR (H.C.F) OR GREATEST COMMON FACTOR (G.C.F.)
The common factor which is highest among the commonfactors of two or more than two numbers is called H.C.F.of that numbers.
Methods to find H.C.F. :
(i) By listing factors:
Factors of 12 = 1. 2. 3. 4. 6, 12
Factors of 32 = 1. 2.4. 8,16, 32
Common factors of 12 and 32 = 1.2,4
H.C.F. of 12 and 32 = 4
(ii) Division Method: This method is useful to find the H.C.F. of greater numbers. This method involves the following steps:
(d) The last divisor is the H.C.F.
Example: Find the HCF of 762 and 1270.
So, HCF of 762 and 1270 is 254.
(iii) By prime factorization method:
H.C.F. of 12 and 24
\[12=2\times 2\times 3\]
\[24=2\times 2\times 2\times 3\]
Common prime factors \[=2\times 2\times 3=12.\]
So. H.C.F of 12 and 24 = 12.
It is Diwali night, Pinki & her family are celebrating & enjoying together. Pinki saw the lightning’s all around.
She saw many colorful lights. She started counting thenumber of bulbs.
She saw that blue bulbs are placed at 4th place, 8th place, and 12th place and so on.....
Let us observe these numbers. 4, 8, 12......
This is similar to the table of 4 or we can say that thesenumbers are the multiples of 4. Can you tell the nextmultiple of 4?
DEFINITIION OF MULTIPLES
For example: If A x B = C, C is multiple of both Aand B.
For Example, let us find out the multiples of 3 & 5.
Multiples of 3: 3, 6, 9,12,15,18, 21, 24....
Multiples of 5: 5,10,15, 20, 25.......
Therefore, the common multiple of 3 <& 5 is 15.
Least Common Multiple (L.C.M.)
The least common multiple among the common multiples of two or more than two numbers is called their L.C.M.
Methods to find L.C.M.
(i) Listing multiples and then finding out the least common multiple.
Example : Multiples of 5 = 5,10,15, 20, 25, 30, 35, 40, ———
Multiples of 7 = 7, 14, 21, 28, 35,42, -——
Common multiples of 5 and 7 = 35, 70, ----
Least common multiple of 5 and 7 = 35.
(ii) Prime factorization method (Division method)
Now, \[2\times 2\times 3\times 3\times 2=72\]
Hence, L.C.M. of 36 and 72 = 72.
Relationship between L.C.M. and H.C.F.
\[L.C.M\times H.C.F.=\]Product of the two numbers.
Rules of Divisibility
Note: If a number is divisible by another number, it isalso divisible by each factor of that number.
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