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High School Mathematics Extensions/Primes/Problem Set

Problem SetEdit

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 x2y, 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.