Let a, b, c, d and e be consecutive integers. Use the Division Algorithm to prove that 5

For each of the following pairs m, n of integers, find the quotient q and remainder r when m is divided by n. Then write m = nq + r. (a) m = 48, n = 11. (b) m = 0, n = 11. (c) m = 48, n = 11. (d) m = 9, n = 11.

For each of the following pairs m, n of integers, determine m/n and m nm/n. (a) m = 38, n = 11. (b) m = 22, n = 11. (c) m = 38, n = 11.

Determine m div n and m mod n for the following pairs m, n of integers. (a) m = 47, n = 14. (b) m = 81, n = 22. (c) m = 180, n = 45. (d) m = 24, n = 25. (e) m = 0, n = 15. (f) m = 55, n = 27.

Show that for each integer n 4, at most two of the three integers n, n + 2 and n + 4 are primes.

Show that if n is an odd integer, then n2 has a remainder of 1 when divided by 4.

Give an example of two integers a and b satisfying all three of the conditions below: (1) a mod 4 6= b mod 4, (2) a2 mod 4 6= a mod 4 and (3) b2 mod 4 6= b mod 4.

Do there exist two distinct positive integers a and b such that a mod b = b mod a?

Show that if 3 n, then n2 has a remainder of 1 when divided by 3.

For each integer n, prove that (a) 3 divides one of the integers n, n + 1 and 2n + 1. (b) 3 divides one of the integers n, 2n 1 and 2n + 1.

According to the Division Algorithm, what are the possible ways that an integer n can be expressed when n is divided by 5?

Let a, b, c, d and e be consecutive integers. Use the Division Algorithm to prove that 5 divides one of these five integers.

Prove that 4 always divides the product of every four consecutive integers but never divides the sum of these integers.

Let p be a prime. Prove that (a) p divides the product of every p consecutive integers and (b) with exactly one exception for p, the prime p divides the sum of every p consecutive integers. (Recall that 1 + 2 + + n = n(n+1) 2 for every positive integer n.)

Let n be an integer. Prove that if 3 n, then 3 | (2n2 5).

Let a, b Z. Prove that 4 | (a2 b2) if and only if a and b are of the same parity.

Let a Z. Prove that if 3 a2, then 3 | (a2 1).

Let a, b Z. Prove that if 3 a and 3 b, then 3 | (a2 b2).

Let n Z. Prove that 3 | (2n2 + 1) if and only if 3 n.

Show that, except for 2 and 5, every prime can be expressed as 10k + 1, 10k + 3, 10k + 7 or 10k + 9 for some integer k.

Let p 3 be a prime. Show that p = 4k + 1 or p = 4k + 3 for some integer k.

Let p 5 be a prime. Show that p = 6k + 1 or p = 6k + 5 for some integer k.

Let n Z. Show that if n = 6k + 5 for some integer k, then n = 3 + 2 for some integer .

Let n Z. Prove that if 3 | (n2 + 1), then 3 (2n2 + 1).