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Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.)

There are two methods of finding the H.C.F. of a given set of numbers:

1. Factorization Method

2. Division method

1. Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

2. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F.

Finding the H.C.F. of more than two numbers: Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number.

Similarly, the H.C.F. of more than three numbers may be obtained.

Least Common Multiple (L.C.M.)

The least number which is exactly divisible by each one of the given numbers is called their L.C.M.

There are two methods of finding the L.C.M. of a given set of numbers:

1. Factorization Method,

2. Division Method (Division Method is short cut method of LCM)

1. Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

2. Division Method : Arrange the given numbers in a row in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

Factors and Multiples

if number a divided another number b exactly, we say that a is a factor of b. In this case, b is called a multiple of a.

6. HCF and LCM of Fractions

When solving HCF and LCM questions with fractions these formulas are very helpful

1. H.C.F. = H.C.F. of NumeratorsL.C.M. of Denominators2. L.C.M. = L.C.M. of NumeratorsH.C.F. of Denominators

Problems on Numbers Formulae

1. (a + b)(a - b) = (a2 - b2)

2. (a + b)2 = (a2 + b2 + 2ab)

3. (a - b)2 = (a2 + b2 - 2ab)

4. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

5. (a3 + b3) = (a + b)(a2 - ab + b2)

6. (a3 - b3) = (a - b)(a2 + ab + b2)

7. (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac)

8. When a + b + c = 0, then a3 + b3 + c3 = 3abc.

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Copyright Disclaimer:Section 107 of the Copyright Act Fair Use Contents .

We are forwarding content link(s) from our website to content website & We are not serving any contents. Main Source:Google.com.All the Content PDF link(s) is/are obtained from GoogleSearch for the purpose of Education & Teaching Intention. Not for commercial purpose. Technicalsymposium.com is not liable/responsible for any copyright issues.