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As you've seen, it is easy to tell if a counting number is divisible by 2, 5, or 10 -- just look at the units digit:
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|
Row |
|
Numbers |
|
|
|
1 |
|
1 |
|
11 |
|
21 |
|
31 |
|
151 |
|
2461 |
|
2 |
2 |
12 |
22 |
32 |
152 |
2462 |
|
3 |
3 |
13 |
23 |
33 |
153 |
2463 |
|
4 |
4 |
14 |
24 |
34 |
154 |
2464 |
|
5 |
5 |
15 |
25 |
35 |
155 |
2465 |
|
6 |
6 |
16 |
26 |
36 |
156 |
2466 |
|
7 |
7 |
17 |
27 |
37 |
157 |
2467 |
|
8 |
8 |
18 |
28 |
38 |
158 |
2468 |
|
9 |
9 |
19 |
29 |
39 |
159 |
2469 |
|
10 |
10 |
20 |
30 |
40 |
160 |
2470 |
|
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Blue: Divisible by 2, but not 5 or 10
Red: Divisible by 5, but not 2 or 10
Green: Divisible by 2, 5, and 10
Since 2, 5, and 10 all divide 10 evenly, the divisibility tests for 2, 5 and 10 are similar in that you only have to examine the units digit. If the units digit is 0, then 10 divides the number. If the units digit is 0 or 5, then 5 divides the number. If the units digit is even (0, 2, 4, 6, or 8), then 2 divides the number.
Why does this work? Any multi-digit number can be written as a sum by replacing the units digit with a 0 and adding the original units digit. For example, the five-digit number 12,345 can be written as 12,340 + 5, where 5 is the units digit of the number. The idea that abcde = abcd0 + e can be extended to any number of digits.
The number abcd0 is 10 • abcd, so 2, 5, and 10 all divide the number abcd0. So if the units digit is 0, then 10 divides the number. If the units digit is not 0, then 10 does not divide the number. Similarly, you only need to check the units digit for divisibility by 2 or 5.
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