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Odd Numbers
By Timothy Hele, aged 8 on October 13, 1998:

If you look at the 1,3,7,9,and all the other odd numbered tables but not ones that have 5 as their last digit.
Example the 3 x table upto 10 x: 3,6,9,12,15,18,21,24,27,30.
If you remove the "tens"digits you are left with the following, 3,6,9,2,5,8,1,4,7,0. You now have all the numbers upto ten. The question is why is this so and, why does this not work for the 5x table or any other table with 5 as the last digit as this works for all other odd numbers?

By Simon (sjm78) on October 13, 1998:

This is a very good question and what you say is quite right. It will work for any number that is not in the 2 or 5 times table (however far you go).

important thing here is that 2×5=10, and when you get to a multiple of ten in the times table you are looking at, you are back where you started: the next entry in the times table will have the same last digit as the first one did. In fact, that is the only way the pattern of last digits can repeat itself. If the pattern does not repeat itself in the first ten entries of the times table, then all the last digits must be different and you will get all ten single-digit numbers coming up, as you observed. If it does repeat itself, then you won't get them all.

hat is special about numbers that are in the 2 or 5 times tables (we call them multiples of 2 or 5)? Imagine you have a number that is a multiple of five. When you work out its times table, you multiply it by 1,2,etc. But a multiple of 5 multiplied by 2 is a multiple of ten, so the pattern repeats as soon as you multiply your number by 2. In fact you get the pattern 0,5,0,5,0,5 etc. in thelast digit. In the same way, as soon as you get to multiplying any multiple of 2 by 5, you get a multiple of ten and you get the pattern 0,2,4,6,8,0,2,4,6,8 etc. However, with numbers that aren't multiples of 2 and 5 you can only get a multiple of ten by multiplying by ten, so the pattern will only repeat after ten entries.

Simon