High School Mathematics Extensions/Primes/Problem Set/Solutions

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Question 1 edit

Is there a rule to determine whether a 3-digit number is divisible by 11? If yes, derive that rule.


Let x be a 3-digit number We have




We can conclude a 3-digit number is divisible by 11 if and only if the sum of first and last digit minus the second is divisible by 11.

Question 2 edit

Show that p, p + 2 and p + 4 cannot all be primes. (p a positive integer and is great than 3)


We look at the arithmetic mod 3, then p slotted into one of three categories

1st category
we deduce p is not prime, as it's a multiple of 3
2nd category
so p + 2 is not prime
3rd category
therefore p + 4 is not prime

Therefore p, p + 2 and p + 4 cannot all be primes.

Question 3 edit

Find x



Notice that










Question 4 edit

9. Show that there are no integers x and y such that



Look at the equation mod 5, we have




therefore there does not exist a x such that


Question 5 edit

Let p be a prime number. Show that





E.g. 3! = 1×2×3 = 6

(b) Hence, show that


for p ≡ 1 (mod 4)


a) If p = 2, then it's obvious. So we suppose p is an odd prime. Since p is prime, some deep thought will reveal that every distinct element multiplied by some other element will give 1. Since


we can pair up the inverses (two numbers that multiply to give one), and (p - 1) has itself as an inverse, therefore it's the only element not "eliminated"


as required.

b) From part a)


since p = 4k + 1 for some positive integer k, (p - 1)! has 4k terms


there are an even number of minuses on the right hand side, so


it follows


and finally we note that p = 4k + 1, we can conclude