How to use the “hmm” method to prove to children the principle of divisibility by 9?

   
That night, I talked to Xiaoyin about a book I owned when I was in her grade (third grade): The Won­der­ful 9. It was the only book I ever bought from ele­men­tary school to mid­dle school, so it left a deep impres­sion on me.
 
   
I said that after read­ing this book, I was sur­prised to find that the sum of all nat­ur­al num­bers with two dig­its or more that can be divid­ed by 9 can also be divid­ed by 9. For exam­ple, 9801, 9+8+0+1=18
, because 18 is divis­i­ble by 9, and so is 9801. Fur­ther­more, if the sum of the num­bers can­not be divid­ed by 9, the remain­der is the remain­der of the orig­i­nal num­ber divid­ed by 9. This prin­ci­ple can sim­pli­fy cal­cu­la­tions for many prob­lems.
 
   
My daugh­ter was very curi­ous about this. So we ran­dom­ly thought of some num­bers to test it out, and of course the results came true. I thought we could stop here, but she kept ask­ing me -
 
   
Why is this?
 
   
Oops, if this were a sto­ry, I could tell her, “It’s just a sto­ry.” But this is math, so we have to go to the trou­ble of prov­ing it.
 
   
I took a large piece of white paper (if it was small­er, I prob­a­bly would­n’t be able to cal­cu­late it) and said I would cal­cu­late it for you.
 
   
Let’s start with a four-dig­it num­ber. For exam­ple, let’s say there is a four-dig­it num­ber, and the dig­its of each dig­it are A, B, C, and D.
 
    Assump­tion: A + B
+ C + D = 9M (which means it is divis­i­ble by 9)
 
   
The num­ber is actu­al­ly: 1000A + 100B + 10C + D. Let me try to break it down:
 
    (1+999)A
+ (1+99)B + (1+9)C + D
    = (A + B
+ C + D) + 999A + 99B + 9C
    = 9M +
9(111A + 11B + C)
    = 9(M +
111A + 11B + C)
 
   
How about it? Now you see clear­ly, this num­ber must be divis­i­ble by 9, right?
 
   
I thought I’d explained the math clear­ly enough, but my daugh­ter looked bewil­dered. She said it seemed like it, but she just could­n’t fig­ure out what ABCD meant! Her moth­er then joined in the fun, look­ing at us with a gloat­ing expression—this was the real kick­er; did­n’t you see I was sweat­ing pro­fuse­ly?
 
   
I was real­ly pushed to the lim­it, so I said to Xiaoyin, let’s throw away all ABCD and think of a four-char­ac­ter idiom for me. Xiaoyin hes­i­tat­ed for a moment and blurt­ed out: “Shit and fart”, and the three of us burst out laugh­ing. Okay, “Shit and fart” will work. Let’s look at it again:
 
    Assump­tion: Poop + Shit
+ fart + fart = 9 mmm (this should make it clear­er!)
 
   
This four-dig­it num­ber is: 1000 poops + 100 shits + 10 flat­u­lences + 1 fart (okay, that’s it)
 
    Breaks down to:
    (1+999)
+ (1+99) shit + (1+9) fart +
    = (Poop + Shit
+ fart + 999 shit + 9 fart
    = 9 hmmm +
9 (111 poop + 11 shit + release)
    = 9(hmm +
111 poop + 11 shit + release
 
   
How about it, is it clear enough this time?
 
   
My daugh­ter laughed all the way and said that she final­ly under­stood it. How­ev­er, she real­ly could­n’t hold it in any longer, so she real­ly cov­ered her stom­ach and uri­nat­ed…
 
   
It was the first time I dis­cov­ered that math­e­mat­ics has such mag­i­cal pow­er!
 
Xiaoy­in’s father’s evening diary, Feb­ru­ary 23, 2009