The tests that I have in mind are all the same. Multiply the 1s place by a number that is smaller than 1/2 the prime whose div'ity being tested. Then add (or subtract) to the remaining digits. If the result is div by the prime in question, so is the original number. Here are a few more tests. - ThreadSky

elviszap.bsky.social • 80 days ago

The tests that I have in mind are all the same. Multiply the 1s place by a number that is smaller than 1/2 the prime whose div'ity being tested. Then add (or subtract) to the remaining digits. If the result is div by the prime in question, so is the original number. Here are a few more tests.

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elviszap.bsky.social•80 days ago

To test for divisibility by 23, multiply the 1s place by 7 and add to the priors. So consider 4899. 9x7=63. 489+63=552. Repeat process. 2x7=14, 55+14=69. If you try this again, then 7x9=63, 63+6=69. So you have to know 69=3x23. But also, 7 is 1 mod 3. So you're also testing for div by 3.

elviszap.bsky.social•80 days ago

Hmm. To test for div. by 71, multiply the last digits and subtract it from the priors. Again consider 4899. 489-63=426. 42-6x7=0. Or try 5183. 518-21=497. 49-7x7=0.
From the previous note 4899=69x71. And 23 quarters is $5.75. Test for div by 23? 57+35=92. And 14+9=23.

elviszap.bsky.social•80 days ago

I think I see a pattern. Div by 9: Add the 1s place to the priors. Div. by 11: Subtract the 1s place from the priors. Div by 19: Multiply the 1s place by 2 and add to the priors. Div. by 21: Multiply the 1s place by 2 and sutract from the priors. ...

elviszap.bsky.social•80 days ago

Div by 31: Mult. the 1s place by 3 and sub from the priors. Div by 39: Mult. the 1s place by 4 and add to the priors. Div by 41: Mult. the 1s place by 4 and sub from the priors. Div by 49: Multiply the 1s place by 5 and add to the priors. Div by 51: Multiply the 1s place and sub from the priors.

elviszap.bsky.social•80 days ago

I'll continue in a moment. But note that because of congruences, from 21, 39, 49, and 51 you get tests for 3 and 7, 3 and 13, 7 and 7, and 3 and 17, respectively.

Div. by 59: mult. 1s place by 6 and add to the priors. Div. by 61: mult. 1s place by 6 and sub from priors.

elviszap.bsky.social•80 days ago

Div by 69: mult. 1s place by 7 and add to the priors. Div by 71: mult. 1s place by 7 and sub from the priors. Div. by 79: mult. 1s place by 8 and add to the priors. Div. by 81: mult. 1s place by 8 and sub. from priors. Div by 89: mult. 1s place by 9 and add to the priors. What primes are missing?

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