The following two divisibility tests are widely taught at Dutch schools.
The 9-test. To reduce a number modulo 9, add its decimal digits. E.g., with N = 15098,
Here is another approach to the 11-test. Interpret the number, by pairing the digits, as written in base 100 and start with the 99-test15098 &equiv 1+50+98 = 149 ≡ 1+49 = 50 ≡ 6,
The fact that the last test manipulates only natural numbers could be viewed as a minor advantage. It is more important to see that the 11-test is related to the 99-test in exactly the same way as the 3-test is related to the 9-test. It is now immediately obvious how to reduce N modulo 111, viz. via a reduction modulo 999:15098 ≡ 15+098 = 113 ≡ 2 ,
Two instances of the general method are being taught as isolated tricks. I know that the mathematics involved is absolutely trivial; so much the worse that it isn't taught in all clarity!
Exercise. Prove that none of the decimal numbers 1001, 1001001, 1001001001, 1001001001001, ........ is prime.
|Plataanstraat 5||4 December 1980|
|5671 AL NUENEN||prof.dr.Edsger W.Dijkstra|
|The Netherlands||BURROUGHS Research Fellow.|