Question

1.Given that HCF (135,225)=45.Find LCM (135,225)
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.

Answer 1

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.