High School Mathematics Extensions/Primes/Problem Set

Problem Set 1. Is there a rule to determine whether a 3-digit number is divisible by 11? If so, derive that rule. 2. Show that p, p + 2 and p + 4 cannot all be primes if p is an integer greater than 3. 3. Find x 4. Show that there are no integers x and y such that 5. In modular arithmetic, if for some m, then we can write we say, x is the square root of y mod m. Note that if x satisfies x2 ≡ y, then m - x ≡ -x when squared is also equivalent to y. We consider both x and -x to be square roots of y. Let p be a prime number. Show that (a) where E.g. 3! = 1*2*3 = 6 (b) Hence, show that for p ≡ 1 (mod 4), i.e., show that the above when squared gives one.