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Irrational Theorems Online Maths NCERT Solutions Class 10 Theorem 1 Let P be a prime number and a be a positive integer. If p divides   then show that p divides a. Proof: – Let  p  be a prime number and  a  be a positive integer such that  p  divides We all are aware that every positive integer can be expressed as the product of primes. Let a = p 1  , p 2 …….  p n  where   , ….. , are primes p 1,   p 2 …….  p n a 2  = (p 1 , p 2 …….  p n ) (p 1 , p 2 …….  p n ) a 2  = (p 1 2 , p 2 2  …….  p n 2 ) P  divides a 2 P  is one of  p 1 , p 2 …….  p n P  divides  a Theorem 2 Prove that  √2 is irrational. Proof: – Let  √ 2  be rational and let its simplest form be  a/b Then  a  an  b  are integers having no common factors other than 1 and  b  is not equal  0 √2=a/b 2=a²/b² 2b²=a² 2 divides  a 2  is prime and divides  a ² Let  a=2c  for some integer  c Putting  a=2c  we get 2b²=4c² b²=2c² 2 divide b² 2 divide b 2 is prime and divides b² 2