Take any collection of prime numbers, say 2, 3, 7, 11 and 23. If we multiply some or all of these numbers, allowing them to repeat as many times as we wish, we can produce a large collection of positive integers (In fact, infinitely many). Let us list a few :

1771 = 7 × 11 × 23

8232 = 2 x 2 x 2 x 3 x 7 x 7 x 7

∴ 8232 = 2

The Fundamental Theorem of Arithmetic says that every composite number can be factorized as a product of primes.

It says that given any composite number it can be factorized as a product of prime numbers in a ‘

So, for example, 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2, or any other possible order in which these primes are written. This fact is also stated in the following form:

1) Find the LCM and HCF of 6 and 20 by the prime factorization method.

We have : 6 = 2 × 3

20 = 2 × 2 × 5

20 = 2

Common factors of 6 and 20 are 2

So for HCF take the common number with lowest exponent.

∴ HCF = 2

In LCM take the common factor with highest exponent and the remaining factors.

∴ LCM of 6, 20 = 2

∴ LCM = 4 x 3 x 5 = 60

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2) Find the HCF and LCM of 6, 72 and 120, using the prime factorization method.

6 = 2 × 3,

72 = 2

120 = 2

Common factors are 2, 2

So for HCF take the common number with lowest exponent.

∴ HCF = 2 x 3 = 6

In LCM take the common factor with highest exponent and the remaining factors.

∴ LCM of 6,72 and 120 = 2

∴ LCM = 8 x 9 x 5 = 360.

• Euclid Geometry

• Euclids division lemma

• Euclids division Algorithm

• Fundamental Theorem of Arithmetic

• Finding HCF LCM of positive integers

• Proving Irrationality of Numbers

• Decimal expansion of Rational numbers

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