# High School Mathematics Extensions/Primes/Problem Set

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## Contents |

## Problem Set[edit]

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 *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.