Perfect numbers, coprimes and twin primes

Once you can list the fac­tors of a num­ber and tell a prime from a com­pos­ite num­ber, you start to notice pat­terns among num­bers. Some num­bers have fac­tors that add up in a spe­cial way. Some pairs of num­bers have noth­ing in com­mon. Some primes come in close pairs. Math­e­mati­cians gave these pat­terns names, and you will meet the names again when you sim­plify frac­tions (a frac­tion is in its sim­plest form exactly when its top and bot­tom are coprimes) and when you study primes more deeply.

You already know how to list all the fac­tors of a num­ber. You know that a prime num­ber has just two fac­tors, $1$ and itself. This les­son needs noth­ing new. It gives names to three things you can now spot. One is a num­ber whose fac­tors add up to twice itself. One is a pair of num­bers with noth­ing in com­mon. One is a pair of primes that stand close together. The acad­e­my's sheet gets two of the names wrong. It has Coprimers and Twin primers. The right words are coprimes and twin primes. That is how they are spelt here.

Per­fect num­bers

Add up all the divi­sors (fac­tors) of a num­ber. If the sum is twice the num­ber, it is a per­fect num­ber.

One word here does all the work: all. Every num­ber is a fac­tor of itself. That is why a fac­tor list ends with the num­ber itself. So the num­ber sits inside the sum. That is why the rule asks for twice the num­ber. It does not ask for the num­ber. Take $6$. Its fac­tors are $1, 2, 3, 6$. The $6$ at the end is $6$ itself. Add the rest and you get $6$ again. So the sum is two sixes.

The sheet gives two per­fect num­bers, $6$ and $28$. Here is the check for each, writ­ten out in full.

6

$$\begin{aligned}1 + 2 + 3 + 6 &= 12 \\ &= 2 \times 6\end{aligned}$$

28

$$\begin{aligned}1 + 2 + 4 + 7 + 14 + 28 &= 56 \\ &= 2 \times 28\end{aligned}$$

Now take the num­ber itself out of each sum. Look at what is left. For $6$ that leaves $1 + 2 + 3 = 6$. For $28$ it leaves $1 + 2 + 4 + 7 + 14 = 28$. So a per­fect num­ber is as big as its other fac­tors put together. That says the same thing as the rule. It just says it the other way round.

Most num­bers fail this test. It helps to watch them fail. They miss in both direc­tions.

Worked exam­ple: is 18 per­fect?

List the fac­tors in pairs so that none is missed: $1 \times 18$, $2 \times 9$, $3 \times 6$. So the fac­tors are $1, 2, 3, 6, 9, 18$.

$$\begin{aligned}1 + 2 + 3 + 6 + 9 + 18 &= 39 \\ 2 \times 18 &= 36\end{aligned}$$

The sum is $39$, which is more than $36$, so $18$ is not per­fect. Per­fect num­bers are rare: after $6$ and $28$ the next one is $496$, and the one after that is $8128$.

Remem­ber. A num­ber is per­fect when all its fac­tors add up to twice the num­ber. Count the num­ber itself as a fac­tor. $6$ and $28$ are the two to know.

Coprimes

Two num­bers are coprimes when their only com­mon divi­sor (fac­tor) is $1$. The sheet gives them a sec­ond name as well: rel­a­tively prime. It is just another name for the same thing.

So two num­bers are coprimes when only $1$ goes into both. Three things are worth fix­ing in your mind. The word turns up else­where too. It is easy to hear it wrong. First, it is about a pair. No sin­gle num­ber is coprime on its own. Sec­ond, it does not say that a num­ber is prime. The table below has coprime pairs with no prime in them. Third, the two fac­tor lists give you the answer. Write out both lists. If $1$ is the only num­ber in both, they are coprimes. If any other num­ber is in both, they are not.

Look at any fac­tor list you have ever writ­ten. $1$ is at the front of it. Every num­ber can be divided by $1$. So any two num­bers at all share $1$. Coprime says they share noth­ing else. Write out both lists and hold them side by side. Then $1$ is the only num­ber in both.

The sheet's exam­ples are $4$ and $5$, and $2$ and $3$. Four more pairs are added here. They let you watch the test pass and fail.

Two rows are worth a sec­ond look. Nei­ther $9$ nor $10$ is a prime num­ber. Nor is $8$ or $15$. Yet both pairs are coprimes. The word says noth­ing about either num­ber being prime. It says only that the two share no fac­tor. The fail­ing rows show what shar­ing looks like. $6$ and $8$ are both even, so $2$ is in both lists. $14$ and $21$ are both built from $7$.

Worked exam­ple: are 25 and 36 coprimes?

Fac­tors of $25$: $1, 5, 25$. Fac­tors of $36$: $1, 2, 3, 4, 6, 9, 12, 18, 36$. The only num­ber in both lists is $1$, so $25$ and $36$ are coprimes, even though nei­ther is prime.

A quick first check helps. If both num­bers are even, $2$ is shared and they can­not be coprimes. If both end in $0$ or $5$, $5$ is shared. Two num­bers next to each other, such as $9$ and $10$, are always coprimes, because any num­ber that divides both would also have to divide the gap between them, which is $1$.

Remem­ber. Coprime is about a pair, never one num­ber on its own. Write out both fac­tor lists. If $1$ is the only num­ber in both, they are coprimes.

Twin primes

Twin primes are two prime num­bers that dif­fer by $2$.

The sheet writes its exam­ples as pairs in brack­ets. So $(3, 5)$ means the two num­bers $3$ and $5$ taken together. Its exam­ples are $(3, 5)$, $(5, 7)$, $(11, 13)$ and $(41, 43)$. Check the last one. The only fac­tors of $41$ are $1$ and $41$. The only fac­tors of $43$ are $1$ and $43$. So both are prime. And $43 - 41 = 2$, so they are two apart. Both halves of the test have to pass.

Why $2$, and not $1$? Because $2$ is the only even prime. Prime and com­pos­ite num­bers sets that out and explains it. So every prime after $2$ is odd. And two odd num­bers are never one apart.

One pair does stand one apart: $2$ and $3$. Both are prime, because $2$ is the even one. They are the only pair like that. They are still not twin primes. Twins must dif­fer by $2$. Apart from $2$, every prime is odd. So the gap between any two odd primes is $2$ or more. No two primes can sit closer than a twin pair does.

The pairs do not stop at $(41, 43)$. Here are the prime num­bers up to $50$. The sheet has a full list of the $25$ primes below $100$. You will find it in Prime and com­pos­ite num­bers.

Run down that list. For each prime, look for the next one that is only $2$ big­ger. Six pairs turn up below $50$. Carry on as far as $100$ and two more turn up. Those are the last ones below $100$.

Some pairs almost work. They teach you as much as the pairs that do. $(7, 9)$ is two apart, but it fails. The rea­son is that $9$ has a third fac­tor: $9 = 3 \times 3$. Its fac­tors are $1, 3, 9$, so it is not prime. $(13, 15)$ fails in the same way. Here $15 = 3 \times 5$, and the fac­tors are $1, 3, 5, 15$. $(89, 91)$ looks right and still fails. This time $91 = 7 \times 13$. Being two apart is never enough on its own.

Worked exam­ple: are 101 and 103 twin primes?

First the gap: $103 - 101 = 2$. Now test each num­ber for primes up to its square root. Since $11 \times 11 = 121$, it is enough to try $2, 3, 5$ and $7$. Nei­ther num­ber is even. The digit sums are $2$ and $4$, so $3$ does not divide either. Nei­ther ends in $0$ or $5$. And $101 = 7 \times 14 + 3$, $103 = 7 \times 14 + 5$. Both are prime, so $(101, 103)$ is a twin prime pair.

Remem­ber. Twin primes are two primes with just one num­ber between them. Check that both num­bers are prime before you call them twins.

All twin primes are coprimes

The sheet adds one note to the twin primes. It says that all twin primes are coprimes. You do not have to just believe it. The two rules you have just read prove it.

A prime brings only two num­bers to its fac­tor list: $1$ and itself. So the fac­tors of $11$ are $1, 11$. The fac­tors of $13$ are $1, 13$. Now put the two lists side by side. $1$ is in both. $11$ is not in the sec­ond list. That is because $13$ is prime and $11$ is not $13$. $13$ is not in the first list, for the same rea­son. So only $1$ is shared. And that is just what coprime means.

Notice what the proof never used: the gap of $2$. It used only two facts. Both num­bers are prime. And they are two dif­fer­ent num­bers. Twin primes are always two dif­fer­ent primes. So the note holds for every twin pair there will ever be. It is not just about the three pairs above.

Remem­ber. All twin primes are coprimes. Two dif­fer­ent primes have noth­ing to share but $1$.

Your turn

Per­fect or not

Write out all the fac­tors of each num­ber. Add them up. Then com­pare the sum with twice the num­ber.

Coprime or not

Write out both fac­tor lists. Say which pairs are coprimes. Where a pair is not, name every fac­tor they share apart from $1$.

Twin primes or not

Say which of these pairs are twin primes. For a pair that is not, say which half of the test it failed.

1. Show that 59 and 61 are coprimes by writ­ing out both fac­tor lists
2. Find a pair of num­bers which are coprimes although nei­ther of them is prime

Com­mon mis­takes

• Leav­ing the num­ber out of its own fac­tor list. Then you must com­pare the sum with the num­ber, not with twice the num­ber. Mix­ing the two meth­ods gives wrong answers.
• Call­ing one num­ber coprime. Coprime always describes a pair, such as $8$ and $15$.
• Think­ing coprimes must be prime. $9$ and $16$ are coprimes, and nei­ther is prime.
• Check­ing only the gap for twin primes. $(37, 39)$ is two apart, but $39 = 3 \times 13$, so it is not a twin pair.
• Call­ing (2, 3) twin primes. They are one apart, not two apart.
• Miss­ing a fac­tor. List fac­tors in pairs, $1 \times 36$, $2 \times 18$ and so on, until the pairs meet.

Key terms

Fac­tor (divi­sor)

A num­ber that divides another exactly. The fac­tors of $6$ are $1, 2, 3, 6$.

Prime num­ber

A num­ber with exactly two fac­tors, $1$ and itself.

Com­pos­ite num­ber

A num­ber with more than two fac­tors, such as $9$.

Per­fect num­ber

A num­ber whose fac­tors, itself included, add up to twice the num­ber.

Coprimes (rel­a­tively prime)

Two num­bers whose only com­mon fac­tor is $1$.

Twin primes

Two prime num­bers that dif­fer by $2$.

Com­mon fac­tor

A num­ber that is a fac­tor of both num­bers in a pair.

Answers

Per­fect or not

1. $6$: $1 + 2 + 3 + 6 = 12 = 2 \times 6$. Per­fect.
2. $10$: $1 + 2 + 5 + 10 = 18$, but $2 \times 10 = 20$. Not per­fect; the sum is too small.
3. $16$: $1 + 2 + 4 + 8 + 16 = 31$, but $2 \times 16 = 32$. Not per­fect; the sum is too small.
4. $20$: $1 + 2 + 4 + 5 + 10 + 20 = 42$, but $2 \times 20 = 40$. Not per­fect; the sum is too big.
5. $28$: $1 + 2 + 4 + 7 + 14 + 28 = 56 = 2 \times 28$. Per­fect.
6. $36$: $1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 + 36 = 91$, but $2 \times 36 = 72$. Not per­fect; the sum is too big.

Coprime or not

1. $4, 9$: fac­tors $1, 2, 4$ and $1, 3, 9$. Coprimes.
2. $6, 35$: fac­tors $1, 2, 3, 6$ and $1, 5, 7, 35$. Coprimes.
3. $8, 12$: fac­tors $1, 2, 4, 8$ and $1, 2, 3, 4, 6, 12$. Not coprimes; they share $2$ and $4$.
4. $10, 21$: fac­tors $1, 2, 5, 10$ and $1, 3, 7, 21$. Coprimes.
5. $15, 25$: fac­tors $1, 3, 5, 15$ and $1, 5, 25$. Not coprimes; they share $5$.
6. $7, 13$: fac­tors $1, 7$ and $1, 13$. Coprimes.
7. $9, 16$: fac­tors $1, 3, 9$ and $1, 2, 4, 8, 16$. Coprimes.
8. $14, 15$: fac­tors $1, 2, 7, 14$ and $1, 3, 5, 15$. Coprimes.
9. $18, 24$: fac­tors $1, 2, 3, 6, 9, 18$ and $1, 2, 3, 4, 6, 8, 12, 24$. Not coprimes; they share $2$, $3$ and $6$.
10. $5, 20$: fac­tors $1, 5$ and $1, 2, 4, 5, 10, 20$. Not coprimes; they share $5$.

Twin primes or not

1. $5, 7$: twin primes.
2. $7, 9$: not twin primes; $9 = 3 \times 3$ is not prime.
3. $13, 17$: not twin primes; both are prime but they dif­fer by $4$.
4. $17, 19$: twin primes.
5. $23, 25$: not twin primes; $25 = 5 \times 5$ is not prime.
6. $29, 31$: twin primes.
7. $37, 39$: not twin primes; $39 = 3 \times 13$ is not prime.
8. $59, 61$: twin primes.

The last two ques­tions

1. The fac­tors of $59$ are $1, 59$ and the fac­tors of $61$ are $1, 61$. The only num­ber in both lists is $1$, so $59$ and $61$ are coprimes.
2. A model answer: $8$ and $9$. The fac­tors of $8$ are $1, 2, 4, 8$ and the fac­tors of $9$ are $1, 3, 9$; only $1$ is shared, and nei­ther num­ber is prime. Other cor­rect pairs include $9$ and $10$, $8$ and $15$, or $4$ and $9$.