[net.math] interesting division by three property

jcn@aplvax.UUCP (08/24/83)

There are many well known divisibility tests. Here are a few:

A number is divisible by two if the last digit is divisible by 2.

A number is divisible by three if the sum of the digits is divisible by 3.

A number is divisible by four if the last two digits of the number is
divisible by four.

A number is divisible by five if the last digit is 0 or 5.

A number is divisible by six if the sum of the digits is divisible by
three and the number is even (i.e. the number is divisible by both
two and three).

I do not know a divisibility test for seven. I have heard that one exists,
but that it is as difficult to perform as the actual long division.
Anybody know what it is?

A number is divisible by eight if the last three digits of the number are
divisible by eight.

A number is divisible by nine if the sum of the digits is divisible by
nine.

A number is divisible by ten if (surprise! :-)) the last digit is 0.

A number is divisible by eleven if the alternating sum of the digits
is divisible by eleven.

A number is divisible by twelve if the last two digits are divisible by
four and the sum of the digits is divisible by three.


Any others?

A lot of people evidently learned these divisibility rules in elementary
school.  I didn't learn them till I taught them in a freshman math
course.

					John Noble
					JHU/APL
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dje@5941ux.UUCP (08/25/83)

There is a test for divisibility by 7:

An integer of the form 10t+u (u is 0 to 9, t is unbounded) is divisible by 7
if and only if t-2u is also divisible by 7.

Example:  test 6543210 for divisibility by 7.  6543210 -> 654321 -> 65430
-> 6543 -> 648 -> 48 not divisible by 7.

Example:  6543215 -> 654311 -> 65429 -> 6524 -> 644 -> 56 is divisible by 7.

Proof: (10t+u) - 7(t+u) = 3(t-2u).

Footnote:  10t+u is divisible by 17 if and only if t-5u is.

Dave Ellis / Bell Labs, Piscataway NJ
...!{hocda,ihnp4}!houxm!houxf!5941ux!dje
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laura@utcsstat.UUCP (Laura Creighton) (08/26/83)

Note for division by 9 -- just keep adding the digits together. if
you don't get 9, then you have a loser.

laura creighton
utzoo!utcsstat!laura

no5db@ihuxl.UUCP (08/29/83)

	A number is divisible by seven if you can multiply the last digit
in the number by 2, strip that digit off the number and then subtract
the remaining number from the product and get a number divisible by 7.
For example, 154, 4 x 2 = 8, 8 - 15 = -7, 154 is divisible by 7.

				Lance