“Casting out nines”: Wolfram Math: Casting Out Nines -- from Wolfram MathWorld
For a mess of them:
“Divisibility tests”: Wolfram Math: Divisibility Tests -- from Wolfram MathWorld
In RSA-style encryption, you first raise a secret to one power, then undo by raising to another power. But, here, we first raise the secret to some power, then undo by raising another base to that. So not quite the same.
An RSA-style trick would be this:
Pick any three digit number you like. Write a 1 after it and then raise it to the 63rd power. Chop off the last digit from the result, and then tell me the last three digits. (For example, if your original number was 712, it would become 7121[sup]63[/sup], whose last digits are …42419761, and you would tell me 976)
I can now recover your original number by adding back in the final 1 you chopped off, raising it to the 127th power, chopping off the last 1 again, and reporting the last three digits. (For example, 976 becomes 9761[sup]127[/sup], whose last digits are …6061297121, and I would report to you 712)
And there are variants of this for any number of digits, any base, and also allowing different pair of exponents, and even allowing different digits to add and chop off, the latter subject to certain coherence conditions. That’s an RSA-style trick.
But it’s not what the trick in post #37 was.
So, am I going to get anywhere looking at the binomial expansions of (1+A)^1000 and (1+8220)^B ? It started to look hopeless. If not, I give in. Term starts in two days and I’ve done exactly 0 prep.
Binomial expansions are certainly good things to look at. I’d advise (1 + 10A)^10. Indeed, I’d advise just playing around and seeing what happens to numbers ending in 1 as you repeatedly raise them to 10th powers. That should make at least some relevant patterns clear, which may spur further thoughts. I’ll provide a fuller explanation in a few days if there’s still interest.
I’ll also note that the significance of the magic number 8221 is that there is a sense in which e[sup]20[/sup] = …8221 (in base ten), and a sense in which the procedure outlined at the beginning of post #37 carries out the operation ln(1 + 10x)/20 in mod 1000 land.
Ah, there was not much interest, and so I’ll leave it be for future readers to puzzle out.
[Dear future readers: it may be of use to you to more fully note that e[sup]20[/sup] = …27197858457785818756568221.0, while ln(21) = …06843588182236083829935820.0]
Well, I opened this thread just now, hoping to learn the answer! If that doesn’t rise to your definition of “interest,” I’ll understand.
First post in ummmmm a long time.
I like that trick, indistinguishable. I will give it some thought.
While out process serving I get to guess ages all day long, its fun. Sometime dead on but always within 5 years and usually with 3.
Its hard sometimes because you dont want to insult the women
The weight too…pretty good at that one too! On women I always guess less than what they look like…if I feel good that day I will do an obviously 30+ woman and guess 25…that really thrills them! Course I have to do anything to get them back in a good mood or in a good mood after I serve them!
Most dont like getting served. Not too many at all slam the door on me!!