MEI Maths Item of the Month
December 2010
10 days of Christmas
We are used to writing numbers in base 10 (we have digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and use powers
of 10 for the columns when writing out numbers) but is this the most convenient number base? It
corresponds well to our number of fingers but it is less convenient when dividing because 10 only
has 2 factors (other than 1 and itself): 2 and 5.
An alternative is to use base 12. This uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X, E and "twelve" is
written as 10 but often said as "do" (short for dozen) to avoid confusion. 12 has 4 factors (other than
1 and itself): 2, 3, 4 and 6 and so using base 12 can make much arithmetic easier.
Base 12, or dozenal, is not as bizarre as it may first sound - we already use base 12 when counting
hours and for feet and inches.
... and finally some dozenal maths:
If you take any prime number > 3 and square it does the result always end in a 1?
e.g.
52 = 21
E2 = X1
152 = 201
2X12 = 80981
Solution
The final digit of a square number is only affected by final digit of the number being squared. Any
prime number > 3 is odd and not a multiple of 3 so must end in 1, 5, 7 or E (in base-12 numbers
ending in 3 or 9 are multiples of 3).
Squaring each of these you get:
12 = 1
52 = 21
72 = 41
E2 = X1
All of which end in a 1.