High School Mathematics Extensions/Primes/Problem Set
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Problem Set
edit1. 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 x^{2} ≡ 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.