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 April 6th, 2018, 02:09 PM #1 Senior Member   Joined: Dec 2015 From: Earth Posts: 832 Thanks: 113 Math Focus: Elementary Math Sum of digits Let $\displaystyle X$ be sum of digits of $\displaystyle 4444^{4444}$ Let $\displaystyle Y$ be sum of digits of $\displaystyle X$ Evaluate sum of digits of $\displaystyle Y$ Last edited by idontknow; April 6th, 2018 at 02:12 PM. April 6th, 2018, 04:46 PM #2 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039 Ever heard of google? https://www.google.ca/search?source=....0.KCxqWg0Hexk April 6th, 2018, 06:02 PM   #3
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Quote:
 Originally Posted by Denis Ever heard of google?
Well if you are just going to tell him that then why not simply use this?

I think the OP is trying to find out how to do it, not to get something to do it for him/her

Have you used modular arithmetic before? What is 4444^4444 mod 9?

-Dan April 6th, 2018, 06:15 PM #4 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 14,597 Thanks: 1039 Suggested google as a way to learn... Thanks from topsquark April 6th, 2018, 07:08 PM   #5
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Quote:
 Originally Posted by topsquark Well if you are just going to tell him that then why not simply use this? I think the OP is trying to find out how to do it, not to get something to do it for him/her Have you used modular arithmetic before? What is 4444^4444 mod 9? -Dan
Ordinarily I'd agree 100% with you Dan. But not in this case. OP has a consistent history of posing difficult number theory problems out of the blue never showing a trace of work on them. It is more like they are posting brain teasers than actually asking for help.

I don't believe OP is even a student. I'm not sure what their purpose here is. April 6th, 2018, 09:23 PM   #6
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 Originally Posted by romsek Ordinarily I'd agree 100% with you Dan. But not in this case. OP has a consistent history of posing difficult number theory problems out of the blue never showing a trace of work on them. It is more like they are posting brain teasers than actually asking for help. I don't believe OP is even a student. I'm not sure what their purpose here is.
Hmph. I never noticed that about him. I'll have to keep my eyes peeled better then.

-Dan April 7th, 2018, 05:13 AM #7 Senior Member   Joined: Dec 2015 From: Earth Posts: 832 Thanks: 113 Math Focus: Elementary Math Im not sure if im correct , but check below Let $\displaystyle Z$ be sum of digits of $\displaystyle Y$ Since $\displaystyle 4444^{4444} <10000^{4444}$ number of digits of $\displaystyle 4444^{4444}$ is less then 20000 so $\displaystyle X<180000 \Rightarrow (Y,Z) \leq (45,12)$ $\displaystyle 4444 \equiv -2 (mod9)$ and $\displaystyle 2\cdot 8^{1481} \equiv 2(-1)^{1481} \equiv 7(mod9)$ $\displaystyle \Rightarrow 4444^{4444} \equiv 7(mod9)$ $\displaystyle 4444^{4444} \equiv X \equiv Y \equiv Z(mod9)$ From $\displaystyle Z \leq 12$ and $\displaystyle Z \equiv 7(mod9)$ we have $\displaystyle Z=7$ April 7th, 2018, 12:22 PM   #8
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Quote:
 Originally Posted by idontknow Im not sure if im correct , but check below Let $\displaystyle Z$ be sum of digits of $\displaystyle Y$ Since $\displaystyle 4444^{4444} <10000^{4444}$ number of digits of $\displaystyle 4444^{4444}$ is less then 20000 so $\displaystyle X<180000 \Rightarrow (Y,Z) \leq (45,12)$ $\displaystyle 4444 \equiv -2 (mod9)$ and $\displaystyle 2\cdot 8^{1481} \equiv 2(-1)^{1481} \equiv 7(mod9)$ $\displaystyle \Rightarrow 4444^{4444} \equiv 7(mod9)$ $\displaystyle 4444^{4444} \equiv X \equiv Y \equiv Z(mod9)$ From $\displaystyle Z \leq 12$ and $\displaystyle Z \equiv 7(mod9)$ we have $\displaystyle Z=7$
$x = 4,444^{4,444} < 10,000^{4,444} \implies log_{10}(x) < log_{10}(10,000^{4,444}) \implies$

$log_{10}(x) < 4444 * 4 = 17,776.$

This does not imply that $x = 4,444^{4,444} < 180,000 \ \because \ 4,444^2 > 180,000.$

It does imply that $y \le 159,984 \implies z < 54.$

But I do not see how that tells you what z actually is.

Last edited by JeffM1; April 7th, 2018 at 12:49 PM. April 10th, 2018, 01:05 AM #9 Banned Camp   Joined: Apr 2016 From: Australia Posts: 244 Thanks: 29 Math Focus: horses,cash me outside how bow dah, trash doves Legendre's formula for the p-adic valuation of a number can be restated in terms of the sum of the digits of the number's p-adic expansion. This would be a starting point I would use for an rigorous determination of an answer to a question regarding the sum of the digits of numbers. https://en.wikipedia.org/wiki/Legendre%27s_formula Tags digits, sum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Gahaja Number Theory 5 May 24th, 2015 08:11 AM zaidalyafey New Users 12 January 18th, 2014 09:22 AM CarpeDiem Elementary Math 7 July 13th, 2012 08:35 PM BuhRock Algebra 6 February 12th, 2011 10:17 AM

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