Category : 5th Class
FACTOR AND MULTIPLE
FUNDAMENTALS
Properties of Factors
Example:
\[85=1\times 85\]
\[73=1\times ~73\]
\[10=1\times 10\] etc.
Example:
\[18=18\times 1\]
\[21=21\times ~1\]
\[15=15\times ~1\] etc.
Example:
\[18=1\times 2\times 3\times 3\] (1, 2, 3, 6, 9, 18 are factor)
\[17=1\times ~17\] (1 and 17 are factors)
\[33=1\times ~3\times ~11\] (1, 3, 11 and 33 are factors)
\[13=1\times ~13\] etc. (1, 13)
Example:
\[2=1\times ~2\] \[(1,\,\,2)\]
\[6=1\times ~6\] \[(1,\,\,2,\,\,3,\,\,6)\]
\[7=1\times ~7\] etc. \[(1,\,\,7)\]
Example: \[0=1,\,\,2,\,\,3,\,\,4...............\]
\[\left( \because \frac{0}{1}=0,\,\,\frac{0}{2}0,\,\,\frac{0}{1}=......... \right)\]
Prime Factorization
Examples:
3 
45 
3 
15 

5 
\[\therefore 45=3\times 5\times 5\]
2 
144 
2 
72 
2 
36 
2 
18 
2 
9 

3 
\[\therefore 144=2\times 2\times 2\times 2\times 3\times 3\]
Highest Common Factor (H.C.F.)
To find the HCF of given number proceed as follows:
Examples: H.C.F. of 15 and 35
\[15=3\times 5\]
\[35=5\times 7\]
Common factor of 15 and 35 is 5.
Hence HCF of 15 and \[35=5\]
H.C.F. by common Davison method
Example: HCF of 14 and 35 by method of Division
Thus HCF of 14 and 35 in 7.
H.C.F. of more than two numbers
Example: HCF of 6, 28 and 44
So, the require HCF is 2
Multiple: A multiple of a number is a obtained by multiplying it by a natural number.
Examples: Multiplies of 36 are \[36\times 1=36,\,\,36\]\[\times 2=72,\,\,36\times 3=144\ldots .\] And so on
Properties of Multiples
Example:
\[3=1\times 3\] \[4=1\times 4\]
\[15=1\times 15\] \[17=1\times 17\]
Example: Multiple of 3 are 3, 6, 9, 12, each of them is greater than equal to 3.
Least common multiple
To find the LCM of two or more numbers
Example: LCM of 6, 14, 7, 32
\[6=2\times 3\]
\[14=2\times 7\]
\[7=1\times 7\]
\[32=2\times 2\times 2\times 2\times 2\]
Prime factors of the given numbers are 2, 3, 7 and their highest powers are\[{{2}^{5}},\,\,3\] and \[7\]
Hence, LCM \[=2\times 5\times 3\times 7=672\]
LCM by short division method
2 
16, 24, 36, 54 
2 
8, 12, 18, 27 
2 
4, 6, 9, 27 
3 
2, 3, 9, 27 
3 
2, 1, 3,9 

2, 1, 1,3 
\[\therefore \,\,LCM=2\times 2\times 2\times 3\times 3\times 2\times 3=432\]
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