- Euclid's Division Lemma states that given positive integer a and b, there exist unique integers q and r satisfying a = bq + r ; 0
 r < b. - Algorithm is a series of well defined steps to solve a problem.
- Euclid's division Algorithm is used to find the HCF of two numbers by aplying Euclid's division lemma to two numbers as follows:
1. Apply Euclid 's Lemma to number a and b with a > b as
a = bq + r ; 0 r < b.
2. If r = 0 then b is the HCF. If r 0 Continue step 1 with a = b and b = r till remainder = 0 . - Fundamental theorem of arithmetic states that 'Every positive composite number can be expressed as the product of primes, uniquely except the order in which the prime factors occur. This theorem is also known as the 'Unique Factorisation Theorem' .
- HCF(a,b)
 LCM(a,b) = a b and HCF(a,b,c) LCM(a,b,c) a bc. - If p is a prime and p divides a2 then p divides a, where a is a positive integer.
- Then decimal expansion of a =
 will terminate if the prime factors of q are of the form 2n5m where n and m are integers and if q is not of the form 2n5m then the decimal expansion will be non-terminating and repeating.
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