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2.State Euclid's Division Lemma and hence find HCF of 16 and 28. 3.State fundamental theorem of Arithmetic and hence Check whether 5×6×2×3+3is a composite number. 4.Prove that 1/2−√5 is irrational number. 5.If the HCF of 210 and 55 is expressible in the form 210×5+55y then find y.

1.135*225=HCF*LCM 135*225=45*LCM 135*225/45=LCM 3*225=LCM 675=LCM or LCM=675 2.Euclid's division lemma-Given positive integers a and b , there exist unique integers q and r satisfying a=bq+r,0 less than or equal to r <b. HCF(16, 28) 16=2*2*2*2 28=2*2*7 HCF=2 3. Fundamental theorem of arithmetic-Every composite number can be expressed (factorised) as aproduct of primes, and this factorisation is unique, apart from the order in which the prime factors occur... 5*6*2*3+3=183 which is a composite number . 4.1/2-√5=a/b therefore, 1/2-a/b=√5 rearranging this equation we get√5=1/2-a/b=b-2a/b since a and b are integers , we get 1/2-a/b is rational and so √5 is rational. but this contradicts the fact that √5 is irrational . This contradiction has has arisen because of our incorrect assumption that 1/2-√5 is rational . so we conclude that 1/2-√5 is irrational.