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The largest positive integer which divides two or more integers without any remainder is called Highest Common Factor (HCF). It is the possible factor of respective numbers.
To find H.C.F by Factorization method
At first, we should find the prime factors of the given number, then the product of the common prime factors is the H.C.F of the given numbers. For example:
Find the H.C.F of 30 and 42
Solution,
Here,
30 = 2× 3× 5× 1
42 = 2× 3× 7× 1
∴ H.C.F = 2× 3× 1
= 6
To the H.C.F by division method
In this method, we divide the larger number by the smaller one and again the first remainder. So obtained divides the first divisor. The process is continued till the remainder becomes zero. The last divisor for which the remainder becomes zero and it is the H.C.F of the given numbers. For example:
Find the H.C.F of 30 and 42
The lower common multiple is the lowest factor of respective numbers.
To find L.C.M by Factorisation method
At first, the prime factor of the given number are to be found out, then the product of the common prime factors and the remaining prime factors(which are not common) is the L.C.M of the given numbers. For example:
Find the L.C.M of 30 and 42
Here,
30 = 2× 3× 5
42 = 2× 3× 7
L.C.M = 2× 3× 5× 7
= 210
Division method
In this method, the given numbers are arranged in a row and they are successively divided by the least common factors till the quotient are 1 or prime numbers. Then, the product of these prime factors is the L.C.M of the given number. For example:
∴ L.C.M = 2× 3× 5× 5× 7
= 1050
Solution:
Here,
F_{(16)} = {1, 2, 4, 8, 16}
F_{(24)} = {1, 2, 3, 4, 6, 8, 12, 24}
F_{(32) }= {1, 2, 4, 8, 16, 32}
Now,
F_{(16)} ∩ F_{(24)} ∩ F_{(32)} = {1, 2, 4, 8}
∴ H.C.F. of 16, 24 and 32 is 8.
Solution:
Here,
28 ÷ 2 = 14 (remainder is 0)
14 ÷ 2 = 7 (remainder is 0)
Now,
28 = 2 × 2 × 7
42 = 2 × 3 × 7
70 = 2 × 5 × 7
∴ H.C.F = 2 × 7 = 14
Solution:
Here,
28 ÷ 2 = 14 (remainder is 0)
14 ÷ 2 = 7 (remainder is 0)
42 ÷ 2 = 21 (remainder is 0)
21 ÷ 3 = 7 (remainder is 0)
70 ÷ 2 = 35 (remainder is 0)
35 ÷ 5 = 7 (remaindder is 0)
Now,
28 = 2 × 2 × 7
42 = 2 × 3 × 7
70 = 2 × 5 × 7
∴ H.C.F = 2 × 7 = 14
Solution:
Here,
24 ÷ 2 = 12 (remainder is 0)
12 ÷ 2 = 6 (remainder is 0)
6 ÷ 2 = 3 (remainder is 0)
36 ÷ 2 = 18 (remiander is 0)
18 ÷ 2 = 9 (remainder is 0)
9 ÷ 3 = 3 (remainder is 0)
Now,
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
∴ L.C.M. = 2 × 2 × 3 × 2 × 3 = 72
Solution:
Here,
M_{(4) }= {4, 8, 12, 16, 20, 24, 28, 32, 36, 20, 44, 48, . . . . . . . }
M_{(6)} = {6, 12, 18, 24, 30, 36, 42, 48, 54, 60, . . . . . . . . }
M_{(8) }= {8, 16, 24, 32, 40, 48, 56, 64, 72, 880, . . . . . . }
Now,
M_{(4) }∩ M_{(6)} ∩ M_{(8) }= (24, 48, . . . . . . . }
∴ L.C.M. of 4, 6, 8 is 24.
Solution:
24 ÷ 2 = 12 (remainder is 0)
12 ÷ 2 = 6 (remainder is 0)
6 ÷ 2 = 3 (remainder is 0)
36 ÷ 2 = 18 (remainder is 0)
18 ÷ 2 = 9 (remainder is 0)
9 ÷ 3 = 3 (renmainder is 0)
48 ÷ 2 = 24 (remiander is 0)
24 ÷ 2 = 12 (remainder is 0)
12 ÷ 2 = 6 (remainder is 0)
6 ÷ 2 = 3 (remiander is 0)
Now,
24 = 2 × 2 × 2 × 3
36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3
∴ L.C.M. = 2 × 2 × 2 × 3 × 3 × 2 = 144
The largest positive integer which divides two or more integers without any remainder is called ______.
The lowest factors of a respective number is known as ______.
The H.C.F. of 20 and 100 is ______.
The H.C.F. of 12 and 60 is ______.
Find the least number which is exactly divisible by 32, 48 and 64.
Find the L.C.M. of 18, 27.
Find the H.C.F. of 60, 90 and 120.
Find the greatest number that divides 60 and 84 without leaving a remainder.
Find the L.C.M. of 5, 50, 75 and 100.
Find the 30, 40, 60, and 80.
ASK ANY QUESTION ON HCF and LCM
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