EWD766

An educational stupidity.

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,

15098 ≡ 8+9+0+5+1 = 23 ≡ 3+2 = 5.

It is proved by observing that, modulo 9, 10 ≡ 1, hence 10^k ≡ 1. As a derivative, we are taught the 3-test: N mod 3 = 2 because 5 mod 3 = 2.

The 11-test. To reduce a number modulo 11, add its decimal digits with alternating signs, e.g.

15098 ≡ 8 - 9 + 0 - 5 + 1 = -5 ≡ 6.

It is proved by observing that, modulo 11, 10 ≡ -1, hence 10^k ≡ (-1)^k.

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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-test

15098 &equiv 1+50+98 = 149 ≡ 1+49 = 50 ≡ 6,

where the last congruence is only modulo 11.

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  ,

where the last congruence is only modulo 111. The possibility of this generalization is well-hidden in the traditional formulation of the 11-test.

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.

transcribed by Om P. Damani
last revision Thu, 8 Nov 2007