Squares and Sums

[CanadaAotS]

I noticed a pattern when you would square a number and add the digits together...

 

ex. 15*15 = 225

2 + 2 + 5 = 9

 

1 - 1

2 - 4

3 - 9

4 - 7

5 - 7

6 - 9

7 - 4

8 - 1

9 - 9

10 - 1

11 - 4

12 - 9

13 - 7

14 - 7

15 - 9

16 - 4

17 - 1

18 - 9

19 - 1

20 - 4

 

I won't go any farther, and just assume that it keeps going.

 

Does anyone know why though? as in a proof as to why?

[NMajik]

I'm sorry if I've misunderstood you, but it seems to me that this isn't going to repeat indefinitely. For example, when 97 is squared, 679 is the result. 6+7+9=22, which isn’t equal to 1, 4, 7, or 9.

[timo]

I'm sorry if I've misunderstood you, but it seems to me that this isn't going to repeat indefinitely. For example, when 97 is squared, 679 is the result. 6+7+9=22, which isn’t equal to 1, 4, 7, or 9.

But 2+2 = 4. And since his period has a length of 9, you would expect the 7th entry to be there, which in fact is 4.

I wouldn´t be too surprised if that periodic behaviour indeed exists and I don´t think it is too hard to proof it, either.

[NMajik]

Good point. I should have given it some more thought.

[insane_alien]

97 squared is 9409

[NMajik]

...Sure it is.... I knew that.

But the sum is still 22...

 

I feel dumb now

[insane_alien]

679 isn't even a square. man, i've been drinkin and i knew that was way off.

[NMajik]

Haha, yeah. That was pretty terrible.

[shmoe]

Consider what adding the digits together does to your number mod 9.

 

(For your pattern, it should be mentioned that you repeat this 'adding the digits together' operation until you get a single digit)

[Severian]

Yes, it is simply because it is mod 9.

 

Clearly you only have 9 possible outcomes. But then you have a symmetry: I could write 21=2x9+3 giving 21²|mod 9 = 3²=9 or 21=3x9-6 giving 21²|mod 9 = 6²|mod 9=36|mod 9 = 9

 

In other words, 3 and 6 will always give the same, as will 2 and 7, etc. This cuts the possible number of outcomes down to 5. The fact that 3²=9 removes a further one, and they will alternate periodically.

 

So there isn't anything deep after realizing that adding the numbers is taking mod 9.

[Ragib]

haha i know this is a pointless post..but 97^2= 679 lol that is awesome! almost as bad as when i screwed up badly in a test, when i did 100/27 and got 268...i dont know how lol..wat was worse was that the topic was consumer arithmetic, and i used that value for the next 5 questions..i cried...