1. Make a list showing all the ways that a number n can be written as a sum of two perfect squares, for each n from 1 to 100. Report any patterns you can find about the numbers that can or cannot be expressed as such a sum. Do you see any patterns regarding the number of such representations?
2. Show that the union of two countable sets is countable. As a corollary, conclude that the union of any finite number of countable sets is countable. What about the union of a countable number of countable sets? -- either prove that such a union must also be countable, or give an example to show that it need not be.
3. Compute the following: (a) the set of all divisors of 120 (b) gcd(1001,343)
4. Show that if x is any integer, then x3-x is a multiple of 3. (Possible hint: If x itself is not a multiple of 3, it's either one more or one less than a multiple of 3.)
1. Find all solutions x to the congruence x^2+1=0 (mod n), first when n=5, then with n=11, then with n=13, and finally with n=65. If you spot any patterns, test them out with some other values of n and report your observations.
2. Show that the product of any three consecutive integers is a multiple of 6.
3. Suppose that a, b and c are three integers and that a|b, b|c, and c|a. Show that a must equal either b or -b.
4. Show that for any integer k, the integers 3k+5 and 5k+8 are coprime. (Hint: any divisor they have in common would have to be a factor of many other integers too...)