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Exercise 1. Prove the following for all positive integers n: a) if n is odd then 8 | n 2 − 1, b) if 3̸ | n and n is odd then 24 | n 2 − 1. Hint: 24 = 3 · 8 and gcd(3, 8) = 1.

Exercise 2. Find some integer solution (where possible): a) 3x − 5y = 7 b) 21x − 35y = 24 c) 97x + 127y = 1

Exercise 3. Simplify the fraction 260 712 561 752

Exercise 4. A number l is called a common multiple of m and n if both m and n divide l. There are many such l. The smallest positive one is called the least common multiple of m and n and is denoted by lcm(m, n). For example 30 = lcm(10, 6) because 30 = 3 · 10 = 5 · 6 (so it’s a common multiple), and any smaller multiple of 10 is not a multiple of 6.

(a) Find lcm(8, 12), lcm(20, 30), lcm(51, 68), lcm(23, 18).

(b) Compare the value of lcm(m, n) with the values of m, n and gcd(m, n). In what way are they related? No need to prove this. Just describe it.

(c) Compute lcm(301337, 307829) using the formula you found in (b). You probably need a calculator for this.

Exercise 5. What is the last digit of 72023?

Exercise 6. Find all integer solutions (x, y) to 6x − 13y = 5