### I'm Eight

Find a great variety of ways of asking questions which make 8.

### Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

### Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

##### Stage: 3 Challenge Level:

It's interesting to see lots of different ways of doing the same calculation - different people seem to prefer different methods, and this was just a handful of potential ways of doing this sum.

Holly, from Hymers College, gave some great explanations of the different methods:

The grid multiplication method works by splitting up the numbers (e.g. 23 into 20 and 3) to times them together more easily (as it is much easier trying to do 20x20 than 23x21). After you times every digit in the first number by every digit in the second number, you simply add them all up.

Column multiplication works in a similar way to the grid multiplication method, as the column method splits the numbers up in a similar fashion.

With the sum 23x21:
you do 1x3, and put the answer in the units column,
then 1x2, and put the answer in the tens column (you are basically doing 1x20 and carrying over from the units to the tens).
For the next line, as you are really doing 20x3, you put a 0 in the units column, and multiply 2x3; the 6 in the tens column represents 60.

The line method works so that after you have drawn all your lines and added them up, they are split into units, tens, hundreds, and so on. This is done by multiplying the units by units (the only way to get units), the tens by units and units by tens (the only ways to get tens), the units by hundreds and the tens by tens (the only ways to get hundreds), and so forth for any
other columns.

The Gelosia method still works with the units x units, tens x units, etc. system explained for the line method of multiplication. The squares are halved diagonally in case you need to 'carry' a digit over to the next column; it will simply be added in with the next column automatically, instead of you having to carry it over yourself.

Alexander from Wilson's School also gave similar explanations, and added his thoughts:

There are a few disadvantages to each of the methods:
Firstly, in Grid Multiplication, you have to do many multiplication sums, and then add them all together.
In column multiplication, the problem is that if you forget to add the zero in the second row, your product will be incorrect.
Line multiplication takes a long time to draw and can get very confusing.
Gelosia multiplication's disadvantage is that it takes a long time to draw, and if you've got a big sum, you have to carry a lot of numbers.

However, all of the methods have advantages:
Grid Multiplication method is very simple, therefore very difficult to get wrong.
Column multiplication is good because it doesn't take up that much space and is very quick.
In Multiplying with lines you don't have to multiply, you just count.
Gelosia multiplication is good because it carries most of the numbers that have to be carried for you.

Chloe from Landau Fort Academy performed the same calculations on a different sum to make sure she had understood the methods, and added her thoughts on what she had done:

Grid method: easy, but takes ages to draw.

Column method: easy, but might get mixed up.

Multiplying with lines: doesn't take long once drawn, but it's quite hard!

Gelosia method: makes sense, but you might mix it up, and it takes a while to draw.

Three students from Westfield Middle School sent us their calculations: Ayesha seemed to like the grid method, Mohammed preferred the Gelosia method and Valentins liked the column method most. Interesting!

Rachel from NLCS Jeju spotted the following clever trick to one of the multiplications:

$23$ is $(22+1)$, and $21$ is $(22-1)$,
so $23\times 21 = (22+1)(22-1)$.

But we know that $(a+b)(a-b) = a^2 - b^2$,
so $23\times 21 = 22^2 - 1 = 484 - 1 = 483$.