High School Mathematics Extensions/Primes/Problem Set/Solutions

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High School Mathematics Extensions75% developed

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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.

Solution

Let x be a 3-digit number We have

now

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)

Solution

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

Solution

Notice that

.

Then

.

Likewise,

and

.

Then

Question 4[edit]

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

Solution

Look at the equation mod 5, we have

but

therefore there does not exist a x such that

Question 5[edit]

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)

Solution

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