Making a powerful difference

9th December 2005 at 00:00
Q: One of my Year 10 asked me what a powerful number was. Can you help?

I like your question because powerful numbers provide an opportunity for pupils to play with numbers.

A: In maths, a number is a powerful number if it is a positive integer that can be exactly divided by a prime number and the prime's square. For example, the powerful number 108 can be divided by the prime number two and by its square (four) to give 54 and 27.

Pupils might like to investigate the fact that powerful numbers can also be written in the form a2 b3, where a, b 3

1. For example, 8 = 12 x 23, and 108 22 x 33. Reducing the numbers to their prime factors first might be useful.

The beginning of the series of powerful numbers (1, 4, 8, 9, 16, 25, 27...) can be seen at primes.utm.eduglossarypage.php?sort=PowerfulNumber An accountant recently asked me if I knew about powerful numbers. I felt super-intelligent providing the above answer. He told me that 9 was a powerful number in accountancy - but in a different way. He said that 9 is used to find errors in calculations and showed me why. Illumination!

I've used an investigation in which you take a number, write the digits in reverse order and subtract. The result is that you always get a number with a digit sum of 9 (that is when the digits of the answer are added together until a single digit is obtained, as in the example). I had never realised that this had an application in accountancy.

With the arrival of computers, it may not be used as much as it used to be, but it does help accountants find the source of mistakes. Quite often, they have to transpose a set of figures from one spreadsheet to another. When a mistake is caused by two digits being reversed, the difference between the sum of the new column of figures and that of the old produces an answer that is divisible by 9. The accountant then knows that they are looking for a reversal of digits mistake.

In the first example, the sum of the original column of figures is Pounds 165.62. When pound;23.49 is incorrectly written as pound;32.49, the total of the new column comes to pound;174.62. Subtracting these gives the answer 9.00.

In the second example, pound;75.10 has been reversed to pound;57.01. This time, the total is pound;147.53. Subtracting pound;165.62 - pound;147.53 gives 18.09, which is divisible by 9 (18.09 9 = 2.01).

If the answer is not divisible by 9, the accountant needs to look for a different type of error. This provides a real application of what is a simple investigation.

Q: I am the maths co-ordinator in a primary school. At a recent Year 3 parents' evening a common question from parents was how they can help their children with numbers. Have you any suggestions?

A: The Basic Skills Agency has some excellent yet inexpensive materials that parents can purchase. Your school could buy some in to offer to parents at evenings such as this.

Go to their website at and look under publications.

The ones I suggest you look at are Talking Numbers with 7 and 8 Year-olds, Digit's Dictionary, and the excellent Detective Digits Calendar, which will have all the family thinking as well as being an excellent classroom resource.

The Basic Skills Agency has other publications that your colleagues might find useful.

Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses.

Email your questions to Mathagony Aunt at Or write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX

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