Conversation

Notices

1. Embed this notice
   Mark Dominus (mjd@mathstodon.xyz)'s status on Friday, 13-Dec-2024 01:51:09 JST Mark Dominus

   Interesting thought this morning. It started with: is 509 prime? So I started doing trial division. Dividing by 13 I found \(509 = 13\cdot 30 +119\) and 119 is not a multiple of 13, okay.

   Now moving on to 17, I can quit immediately *without* dividing because I just saw that \(509 = 13\cdot 30 +119\) and 119 *is* a multiple of 17.

   • Embed this notice
     Mark Dominus (mjd@mathstodon.xyz)'s status on Friday, 13-Dec-2024 01:50:47 JST Mark Dominus

     in reply to

     The overarching trick here is (a) "short division" and (b) short division is usually taught left to right, but it works just as well right to left and you should use the one that seems easier.

     To do \(5486\div 13\) it's easiest to go left to right: \[ 5486 \Rightarrow 286 \Rightarrow 26 \] aha we are done.

     But to do \(4726\div 13\) it's much easier to to right to left \[ 4726 \Rightarrow 47\ne 39\] aha we are done.

     alcinnz repeated this.
   • Embed this notice
     Mark Dominus (mjd@mathstodon.xyz)'s status on Friday, 13-Dec-2024 01:50:53 JST Mark Dominus

     in reply to

     An elaboration on this is that when factoring smallish numbers of the form \(n = 10k+1\) you will often need to test for divisibility by 7 and then by 13 in succession. But \( 7\cdot 13 = 91\) so instead of considering \( n\), consider \[𝑛−91=10k−90=10\cdot (k-9)\] and do both at once.

     For example, instead of dividing \(5031\) by \(7\) and then by \(13\), consider \(503-9 = 494\) instead. This is evidently not divisible by \( 7\), but for \(13 \) you can see from \( 494 + 26 = 520 \) that it is a multiple of \(13\), and therefore so too was \(5031\).

   • Embed this notice
     Mark Dominus (mjd@mathstodon.xyz)'s status on Friday, 13-Dec-2024 01:51:00 JST Mark Dominus

     in reply to

     Another handy trick is, when checking if 509 is a multiple of 19, you do *not* start by dividing 509 by 19 getting 380 plus some yucky remainder. Instead you observe that 509 = 19 + 490 and 49 is not a multiple of 19 so you move on.

     I only thought of this embarrassingly late in life.

   • Embed this notice
     Mark Dominus (mjd@mathstodon.xyz)'s status on Friday, 13-Dec-2024 01:51:16 JST Mark Dominus

     in reply to

     My kids' schools never taught them short division, which seems to me a sad and odd omission.

     The idea is that instead of doing the big hairy thing on the left you do the compact speedy thing on the right. The left-side technique should only be used when the divisor is bigger than the multiplication tables you have memorized.

     The thing on the right is simple enough to do in your head, and particularly so if you only need the remainder and not the quotient. My kids were often mystified by my ability to divide in my head but it's just that I was taught a better algorithm than they were.

     I blogged about this a while back as part of a longer article about several methods of checking for divisibility by 7: https://blog.plover.com/math/divisibility-by-7.html