Prime practice makes perfect

25th February 2005 at 00:00
Q) What other numbers are there that have factors that when added together make the number? I know this works for 6, the factors of six are 1, 2, 3,which add together to make 6. Are there any more?

A) The kind of number you describe is a perfect number. Note that the number itself is not included in the sum. The first three perfect numbers are 6, 28 and 496. Factors of 28 are 1, 2, 4, 7, and 14 (sum = 28). Factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124 and 248 (sum = 496). These were known in the time of Pythagoras, about 500bc. Mathematicians have always been fascinated by patterns in number. Pupils can enter their own voyage of discovery. Euclid, around 300bc, had a method for working them out, and it is suggested that the Egyptians, too, probably knew about them.

First you need to work out 2 to various powers: (20 =1), (21 = 2), (22 = 4), (23 = 8); (20 + 21 + 22 +23 = 16); (25 = 32). The "power" tells us how many times to multiply the number by itself. So 24 = 2 X 2 X 2 X 2 = 16.

Euclid's method involves taking consecutive powers of 2 and adding them together (another way of looking at this is to double each number and add then add it) until you reach a prime number. Then multiply the prime by the last value of the power sequence. This is illustrated in the table below.

There are of course "amicable", "sociable", and many other kinds of numbers - another adventure!

Finding the perfect numbers is a way of practising other parts of maths - eg factors and number powers, and, without using a calculator, multiplication and simple addition.

For an extension of this work visit

For an historical perspective

For an algebraic definition http:mathworld.wolfram.comPerfectNumber.htm

Q) I work as part of a team in educational publishing. I visited the tax website the other day and found that on the section were you fill in to pay your tax there is both Pounds and a p sign, when you have paid, just Pounds remains. Should both be used or only one? This has always confused me.

A) I checked the website and found there are boxes beside where you enter the amounts. They are there for direction only, not as a mathematical notation.

When writing a sum of money, either the Pounds sign should be used or the 'p' sign, as these determine the units you are using. For example, pound;3.45 is correct as is pound;345p, whereas pound;3.45p is incorrect.

Confusion sometimes occurs for those who used the old Pounds s d before we decimalised our monetary system. In the old system, if I wanted to write, for example, pound;1.76 or 176p (new pence) I would write each unit (Pounds s d), to give about pound;1 15s 2d.

This poem was inspired by a dance course in Bristol.

Kinaesthetic creation

Today on a visit to St Paul's I met a girl who puts up walls A defence against failure

"My name is Hannah"

then her offhand manner

"I'm really not interested!"

The dance timetables are organised

then Hannah's tutors are surprised

"I don't understand time"

"Look there's a clock face there"

they point, she doesn't care

Because it's maths

"There is just a chance If you learnt through dance

It might make sense".

Almost with a sigh

She'll give it a try.

It might help.

Through kinaesthetic creation

Understandings mediation

Time traversed.

Now there's a lady in town

and her defences are down

Mathematics accessed.

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