## March 2021

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In the Math Corner of my March Diary I offered a theorem on divisibility. First I gave a rule for extracting a particular integer
*k*, which might be either positive or negative, from any prime number *p*:

RULE: Any primepgreater than 2 ends with either 1, 3, 7, or 9.

Ifpends with 1 or 9, it's right next to a multiple of 10. It's either 10k+1 or 10k−1. Thatkwill be your multiplier, and you'll sign it with the sign in the previous sentence. So ifpends with 1,kis positive; ifpends with 9,kis negative.

Ifpends with 3 or 7, just multiply it by 3 before applying all that. Forp= 7, tripling gets you 21, sok= +2. Fork= 13, however, you'd be looking at 39, sok= −4.

With that rule established, I stated a theorem, and challenged readers to prove the theorem.

The Stump Divisibility Theorem.

Supposepis a prime greater than 2, and I've extracted from it a signed numberk, according to RULE. SupposeNis any number, with rightmost digitd. Call the rest ofNthe "stump." Thenpdivides exactly intoNiffpdivides into (the stump minusdk).

(I had already told the reader that "iff" is math shorthand for "if and only if.")

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** Proof**: Not much of a stretch. First, by looking at RULE, persuade yourself that

*p*must divide exactly into 10

*k*+ 1. If

*p*= 43, for example, then

*k*works out to −13, so 10

*k*+ 1 is −129, which is 43 × (−3). From the way I've defined

*k*, in fact,

*p*will always divide into 10

*k*+ 1 either ±1 times or ±3 times.

Now decompose *N* into its decimal digits. We know the rightmost digit is *d*; so for some positive whole number *s*
the digits are *a*_{s}, *a*_{s-1},
*a*_{s-2}, …, *a*_{2}, *a*_{1}, *d*, and:

N=a_{s}10^{s}+a_{s-1}10^{s-1}+ … +a_{2}10^{2}+ 10a_{1}+d.

=a_{s}10^{s}+a_{s-1}10^{s-1}+ … +a_{2}10^{2}+ 10a_{1}− 10dk+ 10dk+d.

= 10[(a_{s}10^{s-1}+a_{s-1}10^{s-2}+ … + 10a_{2}+a_{1}) −dk] + 10dk+d.

But what is that thing in the inner parenthesis there? Why, it's the stump! So:

N= 10(stump minusdk) + (10k+ 1)d

Since we know that *p* divides exactly into 10*k* + 1, it divides exactly into one of the other two things,
*N* or 10(stump minus *dk*), iff it divides exactly into the other. QED.