Facts And Formulae:
1.Highest Common Factor:(H.C.F) or Greatest Common Meaure(G.C.M) :
The H.C.F of two or more than two numbers is the greatest number that divides each of them exactly.
There are two methods :
1.Factorization method: Express each one of the given numbers as the product of prime factors.
The product of least powers of common prime factors gives HCF.
Example : Find HCF of 26 * 32*5*74 , 22 *35*52 * 76 , 2*52 *72
Sol: The prime numbers given common numbers are 2,5,7
Therefore HCF is 22 * 5 *72 .
2.Division Method : Divide the larger number by smaller one. Now divide the divisor by remainder. Repeat the process of dividing preceding number last obtained till zero is obtained as number. The last divisor is HCF.
Example: Find HCF of 513, 1134, 1215
Sol:
1134) 1215(1
1134
----------
81)1134(14
81
-----------
324
324
-----------
0
-----------
HCF of this two numbers is 81.
81)513(6
486
--------
27)81(3
81
-----
0
---
HCF of 81 and 513 is 27.
3.Least common multiple[LCM] : The least number which is divisible by each one of given numbers is LCM.
There are two methods for this:
1.Factorization method :
Resolve each one into product of prime factors. Then LCM is product of highest powers of all factors.
2.Common division method.
Problems:
1.The HCF of 2 numbers is 11 and LCM is 693.If one
of numbers is 77.find other.
Sol: Other number = 11 * 693/77=99.
2.Find largest number of 4 digits divisible by 12,15,18,27
Sol: The largest number is 9999.
LCM of 12,15,18,27 is 540.
on dividing 9999 by 540 we get 279 as remainder.
Therefore
number =9999 – 279 =9720.
3.Find least number which when divided by 20,25,35,40
leaves remainders 14,19,29,34.
Sol:
20–14=6
25-19=6
35-29=6
40-34=6
Therefore number =LCM of (20,25,35,40) - 6=1394
4.The least number which should be added to 2497 so that sum is divisible by 5,6,4,3 ?
Sol: LCM of 5,6,4,3 is 60.
On dividing 2497 by 60 we get 37 as remainder.
Therefore number to added is 60 – 37 =23.
Answer is 23.
5.The least number which is a perfect square and is divisible by each of numbers 16,20,24 is ?
Sol: LCM of 16,20,24 is 240.
2 * 2*2*2*3*5=240
To make it a perfect square multiply by 3 * 5
Therefore 240 * 3 * 5=3600
Answer is 3600.
1.Highest Common Factor:(H.C.F) or Greatest Common Meaure(G.C.M) :
The H.C.F of two or more than two numbers is the greatest number that divides each of them exactly.
There are two methods :
1.Factorization method: Express each one of the given numbers as the product of prime factors.
The product of least powers of common prime factors gives HCF.
Example : Find HCF of 26 * 32*5*74 , 22 *35*52 * 76 , 2*52 *72
Sol: The prime numbers given common numbers are 2,5,7
Therefore HCF is 22 * 5 *72 .
2.Division Method : Divide the larger number by smaller one. Now divide the divisor by remainder. Repeat the process of dividing preceding number last obtained till zero is obtained as number. The last divisor is HCF.
Example: Find HCF of 513, 1134, 1215
Sol:
1134) 1215(1
1134
----------
81)1134(14
81
-----------
324
324
-----------
0
-----------
HCF of this two numbers is 81.
81)513(6
486
--------
27)81(3
81
-----
0
---
HCF of 81 and 513 is 27.
3.Least common multiple[LCM] : The least number which is divisible by each one of given numbers is LCM.
There are two methods for this:
1.Factorization method :
Resolve each one into product of prime factors. Then LCM is product of highest powers of all factors.
2.Common division method.
Problems:
1.The HCF of 2 numbers is 11 and LCM is 693.If one
of numbers is 77.find other.
Sol: Other number = 11 * 693/77=99.
2.Find largest number of 4 digits divisible by 12,15,18,27
Sol: The largest number is 9999.
LCM of 12,15,18,27 is 540.
on dividing 9999 by 540 we get 279 as remainder.
Therefore
number =9999 – 279 =9720.
3.Find least number which when divided by 20,25,35,40
leaves remainders 14,19,29,34.
Sol:
20–14=6
25-19=6
35-29=6
40-34=6
Therefore number =LCM of (20,25,35,40) - 6=1394
4.The least number which should be added to 2497 so that sum is divisible by 5,6,4,3 ?
Sol: LCM of 5,6,4,3 is 60.
On dividing 2497 by 60 we get 37 as remainder.
Therefore number to added is 60 – 37 =23.
Answer is 23.
5.The least number which is a perfect square and is divisible by each of numbers 16,20,24 is ?
Sol: LCM of 16,20,24 is 240.
2 * 2*2*2*3*5=240
To make it a perfect square multiply by 3 * 5
Therefore 240 * 3 * 5=3600
Answer is 3600.
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