Further Examples of Divisibility Criteria

Hello!

I am not specialist in mathematics, but like to play with numbers. Was curious whether divisibility criteria by 7, 11, 13, 17, 19 and other simple numbers can be developed.

Those are some that I come up with.

1. 7: 5·number of hundreds - 2 last digit number should be divided by 7. Example: a) 511 5·5-11=14, 14 can be divided by 7, hence 511 is divisible by 7; b) 1554 5·15-54=21, 21 can be divided by 7, hense 1554 is divisible by 7.

2. 11: 10·number of hundreds - 2 last digit number should be divided by 11. Example: a) 726 10·7-26=44, 44 can be divided by 11, hence 726 is divisible by 11; b) 1221, 10·12-21=99, 99 can be divided by 11, hence 1221 is divisible by 11.

3. 13: 4·number of hundreds - 2 last digit number should be divided by 13. Example: a) 715 4·7-15=13, 13 can be divided by 13, hence 715 is divisible by 13; b) 1573, 4·15-73=-13, -13 can be divided by 13, hence 1573 is divisible by 13.

4. 17: 2·number of hundreds - 2 last digit number should be divided by 17. Example: a) 952 2·9-52=-34, -34 can be divided by 13, hence 952 is divisible by 17; b) 1904, 2·19-04=34, 34 can be divided by 17, hence 1904 is divisible by 17.

5. 19: 7·number of thousands - 3 last digit number should be divided by 19. Example: a) 18962 7·18-962=-836, -836 can be divided by 19, hence 18962 is divisible by 19; b) 9614, 7·9-614=-551, -551 can be divided by 19, hence 9614 is divisible by 19.

6. 19: 14·number of hundreds - 2 last digit number should be divided by 19. Example: a) 836 14·8-36=76, 76 can be divided by 19, hence 836 is divisible by 19; b) 551, 14·5-51=19, 19 can be divided by 19, hence 551 is divisible by 19.

Other divisibility criteria can be also obtained. I understand why these criteria work. Does number theory have mathematical proof or substantiation for the above divisibility criteria. I would greatly appreciate your response.

Thank you.

Isaak Rudman

Dear Isaak:

All your criteria are correct and have the same explanation. For example, let's look into the first one. Write the number A as (100a + b). Thus, b is a 2-digit number formed by the last two digits of A. Your criteria says that

A is divisible by 7 iff (5a - b) is divisible by 7. This is indeed the case. Let B = (5a - b). Then A + B = 105a which is obviously divisible by 7. Therefore,

A + B = 0 (mod 7)

and, in particular, either both are divisible by 7, or both are not.

All the best,
Alexander Bogomolny

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