Talk:06-240/Homework Assignment 4

Divisibility by Prime Number

Pls correct me if I were wrong. The operation of cut away the unit digit is a distraction. If we consider the unit digit, the operation basically is a deduction of a number, and that number is divisible by 7. The whole operation is shown as follow:

86415
105	105/7=21
----
8631
21	21/7=3
---
861
21	21/7=3
--
84
84	84/7=12
-
0	0/7=0

Since it is an operation of series subtraction by multiples of 7, therefore the number we started from is divisible by 7 iff the resulting number is divisible by 7.

Moreover, there is a relationship between the unit digit, 2, and 7. The unit digit multiple by 21(7 $\times$ 3) is equal to the combination of the unit digit with its 2-time as the tenth/hundredth digit.

Unit digit, $x$	$x\times 3\times 7$
0	0
1	21
2	42
3	63
4	84
5	105
6	126
7	147
8	168
9	189

From the table above, I've induced the criterion for divisibility by 17 that is similar operation but the unit digit multiplies by 5 instead of 2. For divisibility by 13, the unit digit multiple by 9. Alright, I think it will be more fun if it's explained by other people. Wongpak 09:38, 5 October 2006 (EDT)