LCM AND HCF FOR BANK EXAMS
Prime Numbers:
A Composite number can be uniquely expressed as a product of prime factors. For example
12 = 2 x 6 = 2 x 2 x 3 = 22 x 31
20 = 4 x 5 = 2 x 2 x 5 = 22 x 5
Every Composite number can be expressed in a similar manner in terms of its prime factors.
Number of Factors
The number of factors of a given composite number N (including 1 and the number itself) which can be resolve into its prime factors as,
N = am x bn x cn...... where a,b,c are prime numbers are (1+m)(1+n)(1+p)......
Example:
Find the total number of factors of 240
Solution:
240 = 2 x 2 x 2 x 2 x 3 x 5 = 24 x 31 x 51
comparing with the standard format for the number N, we obtain a = 2, b = 3, c = 5, m = 4, n = 1, p = 1. The total number of factors of this number including 1 and itself are = (1+m) (1+n) (1+p) .... = (1+4) x (1+1) x (1+1) = 5 x 2 x 2 = 20.
Factors of 240 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240 (Total 20 in numbers)
Sum of Factors:
The sum of factors of the number N ( as defined above ) is given by the formula:
where, a,b,c....,m,n,p... retain the same meaning.
Every Composite number can be expressed in a similar manner in terms of its prime factors.
Number of Factors
The number of factors of a given composite number N (including 1 and the number itself) which can be resolve into its prime factors as,
N = am x bn x cn...... where a,b,c are prime numbers are (1+m)(1+n)(1+p)......
Example:
Find the total number of factors of 240
Solution:
240 = 2 x 2 x 2 x 2 x 3 x 5 = 24 x 31 x 51
comparing with the standard format for the number N, we obtain a = 2, b = 3, c = 5, m = 4, n = 1, p = 1. The total number of factors of this number including 1 and itself are = (1+m) (1+n) (1+p) .... = (1+4) x (1+1) x (1+1) = 5 x 2 x 2 = 20.
Factors of 240 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240 (Total 20 in numbers)
Sum of Factors:
The sum of factors of the number N ( as defined above ) is given by the formula:
(am+1-1)(bn+1-1)(cp+1-1)....
|
(a-1)(b-1)(c-1)............
|
where, a,b,c....,m,n,p... retain the same meaning.
Example: Find the Sum of all factors of 240.
Solution:
The sum by the above formula =
= 744.
We can see that:
1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 16 + 20 + 24 +30 + 40 + 60 + 80 + 120 + 240 = 744
The number of ways in which a composite number N may be resolve two factors
1/2 (p+1)(q+1)(r+1)....... if N = apbqcr is not perfect square and
= 1/2 [(p+1)(q+1)(r+1).......] if N is a perfect square.
The number of ways in which a composite number can be resolved into two factors which are prime to each other. if N= apbqcr...... then the number of ways of resolving N into two factors prime to each other is,
1/2 [(1+1)(1+1)(1+1)....] = 2n -1 where n is the number of different prime factors of N.
If P is a prime number, the coefficient of every term in the expansion of (a+b)p Except the first and the last is divisible by p.
Example: In how many ways can the number 7056 be resolved into two factors?
Sol: N = 7056 = 32 x 24 x 72
Note: N is perfect square. Number of ways in which it can be resolved into two factors.
= 1/2 {(p+1)(q+1)(r+1)+I}
= 1/2 {(2+1)(4+1)(2+1) +1} = 1/2 x 46 =23
HCF of Numbers:
It is the highest common factor of two or more given numbers. It is also called GCF(Greatest Common Factor). For example HCF of 10 & 15 is 5. HCF 0f 100 and 200 is 50.
Factorization method to find HCF:
To find the HCF of given numbers, first resolve the numbers into their prime factors. After expressing the number in terms of the prime factors, the HCF is the product of common factors.
Example: Find the HCF of 88, 24 and 124.
Solution:
88 = 2 x 44=2 x 2 x 22 = 2 x 2 x 2 x 11
24 = 2 x 12= 2 x 2 x 6 =2 x 2 x 2 x 3
124 = 2 x 62 = 2 x 2 x 31
From above three numbers, the common factor is 2 x 2.
HCF = 22.
| (25-1)(32-1)(52-1)... |
| (2-1)(3-1)(5-1)....... |
| 31 x 8 x 24 |
| 1 x 2 x 4 |
= 744.
We can see that:
1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 16 + 20 + 24 +30 + 40 + 60 + 80 + 120 + 240 = 744
The number of ways in which a composite number N may be resolve two factors
1/2 (p+1)(q+1)(r+1)....... if N = apbqcr is not perfect square and
= 1/2 [(p+1)(q+1)(r+1).......] if N is a perfect square.
The number of ways in which a composite number can be resolved into two factors which are prime to each other. if N= apbqcr...... then the number of ways of resolving N into two factors prime to each other is,
1/2 [(1+1)(1+1)(1+1)....] = 2n -1 where n is the number of different prime factors of N.
If P is a prime number, the coefficient of every term in the expansion of (a+b)p Except the first and the last is divisible by p.
Example: In how many ways can the number 7056 be resolved into two factors?
Sol: N = 7056 = 32 x 24 x 72
Note: N is perfect square. Number of ways in which it can be resolved into two factors.
= 1/2 {(p+1)(q+1)(r+1)+I}
= 1/2 {(2+1)(4+1)(2+1) +1} = 1/2 x 46 =23
HCF of Numbers:
It is the highest common factor of two or more given numbers. It is also called GCF(Greatest Common Factor). For example HCF of 10 & 15 is 5. HCF 0f 100 and 200 is 50.
Factorization method to find HCF:
To find the HCF of given numbers, first resolve the numbers into their prime factors. After expressing the number in terms of the prime factors, the HCF is the product of common factors.
Example: Find the HCF of 88, 24 and 124.
Solution:
88 = 2 x 44=2 x 2 x 22 = 2 x 2 x 2 x 11
24 = 2 x 12= 2 x 2 x 6 =2 x 2 x 2 x 3
124 = 2 x 62 = 2 x 2 x 31
From above three numbers, the common factor is 2 x 2.
HCF = 22.
HCF & LCM of Decimals
Example: Calculate the HCF and LCM of 0.6, 0.9, 1.5, 1.2 and 3
Solution: The numbers can be written as 0.6, 0.9, 1.5, 1.2 and 3 consider them as 6, 9, 15, 12, 30
HCF = 3 ⇒ Required HCF = .3 and LCM = 18
Note: If the first number in the above example had been 0.61, then the equivalent integers would have been 61, 90, 150, 120 and 300.
HCF and LCM of Fractions:
HCF of fractions = HCF of numbers ÷ LCM of numbers
Example: Find the HCF and LCM of 5/16, 3/4, and 7/15
Solution:
HCF = HCF(5,3 and 7) / LCM(16, 4 and 15) = 1/240
LCM = LCM (5, 3, 7) / HCF (16, 4 ,15) = 105
Example: Find the HCF and LCM of 5/16, 3/4, and 7/15
Solution:
HCF = HCF(5,3 and 7) / LCM(16, 4 and 15) = 1/240
LCM = LCM (5, 3, 7) / HCF (16, 4 ,15) = 105
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