GROUPS AND RINGS

1. State any two properties of a group. 2. Define Homomorphism of groups. 3. Give an example of Homomorphism of groups. 4. Define Isomorphism.

DIVISIBILITY THEORY AND CANONICAL DECOMPOSITI

1 State divisible algorithm 2 State pigeon hole principle 3 State principle of inclusion and exclusion 6 Express (10110)2 in base 10 and express (1076)10 in base two 7 Express (1776)8 in base 10 and express (676)10 as octagonal 8 Express (1976)16 in base 10 and express (2076)10 as hexadecimal 10 Express (12,15,21) as a linear combination of 12, 15, and 21 12 Use canonical decomposition to Evaluate the GCD of 168 and 180 13 Use canonical decomposition to evaluate LCM of 1050 and 2574 14 Find the canonical decomposition of 2520 15 Find the prime factorization of 420, 135, 1925 16 Using recursion evaluate (252, 360) 17 Using recursion evaluate [24,28,36,40] 18 Using recursion evaluate (18,30,60,75,132) 19 Find the GCD (414,662) using Euclidean algorithm 20 Find the LCM (120.500) 21 State and Prove Euclidean algorithm 22 Find the number of positive integers ≤ 3000 divisible by 3, 5 or 7 23 Prove that the GCD of two positive integers a and b is a linear combination of a and b 24 Find the number of positive integers in the range 1976 through 3776 that are divisible by 13 and not divisible by 17 25 Find the number of integers from 1 to 250 that are divisible by anyof the integers 2,3,5,7 26 Prove by induction that 2𝑛3 + 3𝑛2 + 𝑛 is divisible by 6 for all integers 𝑛 ≥ 0 27 State and prove Fundamental Theorem of Arithmetic. 28 Prove that every integer 𝑛 ≥ 2 has a prime factor. 29 Use Euclidean algorithm to find the GCD of (1819, 3587). Also express the GCD as a linear combination of the given numbers