Sum of digits
No new post? Ok let's do a replay, since Chris taught us how to do this.
The sum of the digits of 173^371 is A. The sum of the digits of A is B. The sum of the digits of B is C. What is the value of C?
The sum of the digits of 173^371 is A. The sum of the digits of A is B. The sum of the digits of B is C. What is the value of C?
Labels: mathschallenge
posted by Ragknot at
8:09 PM
16 Comments:
1^42
Using the same methods as in fffffour, I get 5.
I've just been blown away by the Windows standard issue Calculator, in scientific mode. It can handle 4444^4444 mod 9 directly.
Off topic, but an Anonymous has just completely solved the Tom's Sweet Tooth problem.
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000 for the sweet tooth problem? (Yes, I know that 1^42 is 1, just thought that it would be funnier that way)
Hi Anonymous. I should have acknowledged your humourous answer, sorry about that.
I want to know why the Windows Scientific Calculator can handle 4444^4444 mod 9, but my TI 89 Titanium cant!
Hi Anonymous. I haven't used a TI-89 in many years. I had one of the original one's with a printer cradle and magnetic programming strips (LCDs hadn't been invented then). I thought it was a superbly engineered device. I wrote a program for it that would solve for all roots (real and complex) of a 23rd order real coefficient polynomial (as long as they were sufficiently well behaved).
I guess that the TI-89 only has about 12 display digits and two internal guard digits. The Windows Calculator works to at least tens of thousands of digits internally - but I only found that out a few minutes before my post.
It doesn't say number is too large, exactly, it gives me infinity.
(same anon as above) I just tried 4444^400, it gave me infinity. Then, I tried 4444^40, that gave me the following answer, so it does have more than a 12 digit display:
814,633,022,124,150,824,602,816,740,539,573,392,018,164,750,030,331,198,517,470,434,189,167678,120,281,479,281,634,070,956,306,537,519,860,238,832,557,585,633,178,901,879,111,762,994,136,088,576
By the way, that is without a doubt, the LARGEST exact number I have ever seen.
oh, between the 331 and 120, there is
985,174,704,341,891,167,678
So it would be about 8.14633022 x 10^147
FYI, there are 42 decimal places showing in my last post!
Hi Anonymous. Ooops. I had the TI-59, not the 89. I've just been reading up on the TI-89, the manual says displays 12, but uses 14 internally, normally. It has an EXACT mode for integer arithmetic, and that's good for 614 digits.
www.wolframalpha.com provides a very useful calculator.
That's a miniscule number ;) See http://trickofmind.com/2009/08/grahams-number_21.html
If anyone wants to lern more about modular arithmetic, try this, it's where I learnt it from: http://www.cut-the-knot.org/blue/Modulo.shtml
173^371 has 831 digits. Vista's calculator says
2.0658760497523743860378401054462e+830 which does indicate 831 digits.
173 to the power of 371 has 831 digits
The sum of the 831 digits is 3812 so A= 3812
Summing the digits of A is 14 which is B.
The sum of the digits of B is C which is 5
Below are the 831 digits.
206587604975237438603784010544619995006144884453956834157202452800119579919130307674156661621772276745932559940317961501373790515491901824419835849187461838898651593541424866299589172476264389842092560123501569393889916535739281251864080941177603835104088311988721122293202666607253650612280116651488170847423020157886268001532527689961683289283106853896101710543299719020971399613571082379943941558878697958361736264909724283285532282685394653725266231788654383647548709737717550058833273223860095469160097088074861061333850518091156514653134750390392588395178212438087779671074205437389082434526699964058169415722227683692106353824892524689638754025908947727640602028527911909493445280063928464663451122383136620923365916107341383192377641977499551549140664188234204746412268199674894183899485179665760254859291216460404559975477
Ragknot, LOL everyone would have to trust you on that.
According to the Windows Calculator, 173^731 = 5 (mod 9). 173^731 can also be seen to be 831 digits. So max possible sum of the digits is 831*9 = 7479 ≥ A. Max possible sum of A's digits is 4*9 =33 ≥ B. Max possible sum of Bs digits = 11 ≥ C. So C = 5. Would have had a problem if had found e.g. 20 ≥ C. Then would introduce some statistical guesses such as assuming that the average digit value is 5 (say). Then 831*5 = 4155 ≥ A, => 4*5 = 20 ≥ B => 10 ≥ C.
Chris, I remember the Graham's Number "problem", what I said was that was the largest EXACT number I have seen(of course before Ragknot's post). Once you get a few levels in that power tower, the number does get so massive that it said there is not enough ink in the world to write it by hand. If you could show me where I can actually find the entire number, I would be ecstatic.
Hi Anonymous. I didn't know that you had seen the Graham's number article. Your post was simply gave me an excuse to refer to it. I hoped that you would be amused by it even though it wasn't possible to write it down.
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