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You probably used the test for 2 and the test for 3 to check divisibility by 6. We cannot, however, use the test for 2 twice to check for divisibility by 4. Using 2 and 3 works because they are relatively prime; that is, the only factor they have in common is 1.
The test for divisibility by 2 can be modified for testing divisibility by 4 and 8. Here's how it works.
Since 4 does not divide 10, but 4 does divide 100, rewrite the number, such as the five-digit number abcde, into two parts, abc00 + de; this is 100 times the three-digit number abc plus the two-digit number de. Since 4 divides 100, then 4 divides abc00, and all that needs to be checked is the two-digit number de. If de is divisible by 4, then the entire number is divisible by 4.
For example, to check whether the number 23,456 is divisible by 4, rewrite the number as 23,400 + 56. We know that 4 divides 23,400. Since 4 also divides 56, then 4 divides 23,456.
The test for divisibility by 8 continues this pattern. Because 8 does not evenly divide either 10 or 100, but it does divide 1,000, separate the number -- for example, separate abcde into two parts, ab000 + cde. To test 23,456, write the numbers 23,000 + 456. Since 8 divides 23,000 and 8 divides 456, then 8 evenly divides 23,456.
You can try these tests with the numbers in the chart below:
 |
|
Row |
|
Numbers |
|
|
|
1 |
|
1 |
|
11 |
|
21 |
|
31 |
|
151 |
|
2461 |
|
10,561 |
|
2 |
2 |
12 |
22 |
32 |
152 |
2462 |
10,562 |
|
3 |
3 |
13 |
23 |
33 |
153 |
2463 |
10,563 |
|
4 |
4 |
14 |
24 |
34 |
154 |
2464 |
10,564 |
|
5 |
5 |
15 |
25 |
35 |
155 |
2465 |
10,565 |
|
6 |
6 |
16 |
26 |
36 |
156 |
2466 |
10,566 |
|
7 |
7 |
17 |
27 |
37 |
157 |
2467 |
10,567 |
|
8 |
8 |
18 |
28 |
38 |
158 |
2468 |
10,568 |
|
9 |
9 |
19 |
29 |
39 |
159 |
2469 |
10,569 |
|
10 |
10 |
20 |
30 |
40 |
160 |
2470 |
10,570 |
|
|
Blue: Divisible by 4, but not 8
Red: Divisible by 4 and 8
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