interesting number theory in math
TRANSCRIPT
NUMBER THEORY
HCF AND LCM
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FACTORS AND MULTIPLES
When a number ‘a’ is divided with another number ‘b’
exactly leaving no remainder. There b is factor of a , or
alternatively I can say a is multiple of b. A common factor
is a number which divide them exactly leaving no
remainder.
For example, 2 is factor of 6 and 6 is multiple of 2.
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LCM(LEAST COMMON MULTIPLE)
LCM is the smallest non zero number that is common in two
or
more numbers. It is the smallest number that is multiples of
numbers.
For example:
Multiples of 8 are: 8,16,24,32,……………
Multiples of 24 are: 24,48,72,96,……………..
Therefore, the smallest common multiples between 8 and 24
is 24.
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METHODS TO FIND THE LCM
Following are the methods to find the LCM
Factorization
Division
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FACTORIZATION
In factorization, we will break or factorize the number in prime factors and
them multiply all the prime factors common to all the numbers.
For example:
LCM of 12, 15, 20 is 60.
Factors of 12 are: 2x2x3
Factors of 15 are: 3x5
Factors of 20 are: 2x2x5
So, factors present in all the numbers are: 2x2x3x5=60
Therefore, LCM of 12, 15 and 20 is 60.
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DIVISION METHODIn division method, we divide numbers to their prime factors.
For example: LCM of 12, 27 and 40
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HIGHEST COMMON FACTOR (HCF)
The highest common factor is the largest factor that two
numbers have in common.
Let’s find the GCF of 12 and 42. First, we need to make a
list of factors for each number.
This is also called as greatest common factor (GCF)
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METHODS TO FIND THE HCF
Following are the methods to find the HCF
Factorization
Division
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USING PRIME FACTORIZATION
We need to find the prime factors of each
number and then Identify the common
factors. Then we multiply them together.
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FACTORIZATION METHODEXAMPLE: HCF OF 27 AND 36
27
3 x 9
3 x 3
36
2 x 18
2 x 9
3 x 3
•Both prime factorizations
have 3x3 in common.
•Therefore, the HCF is 9.
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DIVISION METHOD
HCF is the highest common number that is
present in two or more numbers. We can also
find HCF by upside down division.
Example: To find the HCF of 12 and 42
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12
1 x 12
2 x 6
3 x 4
4 x 3
Factors of 12:
1, 2, 3, 4, 6,12
42
1 x 42
2 x 21
3 x 14
4 x ??
5 x ??
6 x 7
7 x 6
Factors of 42:
1, 2, 3, 6, 7, 14, 21,
42
Common Factors: 1, 2, 3, 6
Greatest Common Factor: 6 12© 2 0 1 7 V N A Y A . C O M I N C . A L L R I G H T S R E S E R V E D
WORD PROBLEMS ON LCM AND HCF
1. Judy has 16 red balloons and 24 yellow balloons. She wants to give same
number of each color balloons to kids. What is the greatest number of
Kids she can give balloons?
Answer:
The greatest common number is : 8
So, she can give balloons to 8 kids with same number of red and yellow.
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2. Two traffic lights are turned on at the same time. one light turned on every 9
seconds. the other turned on every 15 seconds. after how many seconds will they on
together again?
Answer:
For the first light: 0, 9, 18, 27, 36, 45, 54, 63,
And for the second light: 0, 15, 30, 45, 60, 75
Therefore , the smallest common number is : 45
So they will turned again after 45 seconds.
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3. Jerusha has 12 oranges, 28 apples, and 36 bananas. what is the greatest number
of baskets she can make if each type of fruit is equally distributed among the
baskets? how many of each supply will go into the baskets?
Answer:
4 baskets with 3 oranges, 7apples, and 9 bananas.
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4.Two clocks are turned on at the same time. one clock chimes every 15 minutes.
the other clock chimes every 25 minutes. in how many minutes will they chime
together?
Answer:
First clock: 0,15,30,45,60,75,90,105,,,,,,,,
And second clock: 0,25,50,75,100,125,,,,,,,
So, smallest common number is : 75
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