Wikipedia:Reference desk/Archives/Mathematics/2022 September 18

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September 18

Common mistakes in divisibility rules

Do people sometimes make mistakes in divisibility rules?? For example, the "add the digits" rule is valid for 3 and 9, but not 11. I believe one common mistake is using the "add the digits" rule for 11. Are there many common mistakes with divisibility rules?? Georgia guy (talk) 18:15, 18 September 2022 (UTC)

I suspect that most people do not even know the divisibility rules for 3 and 9, and I doubt very much that anyone has attempted to produce estimates of the prevalence of such mistakes. It is exceedingly easy to spot that "add the digits" fails for divisibility by 11, since it gives incorrect results for 11, 22, 33, ...; in fact, the least multiple of 11 for which the result is accidentally correct is 209 – and if one keeps adding the digits like in the rule for 3, as in 81752 → 8+1+7+5+2 = 23 → 2+3 = 5, not a single multiple of 11 can be detected. For that reason, I do not think that this specific mistake is common.  --Lambiam 18:55, 18 September 2022 (UTC)

Please note that it's only an example; I'm asking for pretty much any common mistake in divisibility rules that people make. Not just the example I gave; I'm only using it to clear up as an example. Georgia guy (talk) 19:01, 18 September 2022 (UTC)

Well, the mistake in applying "add the digits" when dividing by 11 (aka 191 or -29) is doing it with base 10 arithmetic. Works fine in base -10. :-)John Z (talk) 04:35, 22 September 2022 (UTC)

What is the population you are considering? Obviously, many children who are taught divisibility rules in school without understanding why they work will make all kinds of mistakes when they have to put them in practice, much like for any other maths drill. They may think 678 is divisible by 4 because 8 is divisible by 4. A year later they may have forgotten all these rules, just like all these other maths drill rules. Since they have no need to apply divisibility rules, they will not apply them, whether correctly or incorrectly.  --Lambiam 20:20, 18 September 2022 (UTC)

To be honest, I'm having a hard time seeing this as a math question, more of an education or psychology question. I can tell you if a divisibility rule is correct, but without a classroom full of undergraduates to survey I'd have no way of knowing how many people would get it wrong. Even with a classroom, a lot would depend on how good the teacher is, how well it's covered in the text, and how strongly it's emphasized in class. It would probably also depend on how the question is posed. If you're looking for some kind of national average, then how would that data be collected? Maybe there are experts in educational curricula who could answer this, but I somehow doubt they would be reading this. --RDBury (talk) 08:56, 19 September 2022 (UTC)