Real Numbers

1. Show that every positive even integer is of the form 4q, 4q+2 and every positive odd integer is of the form 4q+1, 4q+3.
   
2. Show that one and only one out of n,n+2 and n+4 is divisible by 3, where n is any positive integer.
   
3. Show that 4n can never end with digit 0.
   
4. Find the HCF and LCM of 10224 and 1608 using prime factorization method and verify the answer using Euclid’s Lemma.
   
5. Find the HCF of 144, 180, 192 using Euclid Algorithm.
   
6. Without actual division, state whether 19/3125 is terminating or non-terminating.
   
7. Prove that √3 is irrational number.
   
8. Prove that √3-√5 is an irrational number.
   
9. Prove that 4-2√5 is an irrational number.
   
10. By Euclid’s division algorithm show that square of any positive integer is of the form 3n or 3n+1.
    
11. By Euclid’s division algorithm show that cube of any positive integer is of the form 9n , 9n+1 or 9n+8.
    
12. Given HCF(306,657)=9, find the lcm of (306,657).