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Topic: A problem on remainders and quotients (Read 732 times) |
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pcbouhid
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A problem on remainders and quotients
« on: Dec 13th, 2005, 10:26am » |
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A number less than one million (1,000,000) is know to be such that diminishing it by 3 makes it divisible by 7. From the number diminished by 3 we subtract the seventh part. We then obtain a number that also becomes divisible by 7 after 3 is subtracted from it. From this number we derive another in the same way, namely by subtracting a seventh part from the number diminished by 3. This time, too, a number results that is divisible by 7 after subtracting 3. This occurs four more times, so that, in all, it happens seven times over, that a number is divisible by 7 after subtracting 3.
a) What is the number?
b) What is the number (that this time lies between 2,000,000 and 3,000,000) if the sixth number formed in the same way as before has the remainder of 5, instead of 3, when divided by 7?
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| « Last Edit: Jan 4th, 2006, 8:10am by pcbouhid » |
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Dis
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #1 on: Dec 13th, 2005, 5:30pm » |
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a) in Java:
class B {
public static void main(String[] args) {
xloop:
for (int x=1; x < 1000000; x++) {
int result = x;
for (int cntr = 0; cntr < 7; cntr++) {
if ((result-3)%7 != 0) continue xloop;
result = 6*(result-3) / 7;
}
System.out.println("x is " + x);
}}}
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| « Last Edit: Dec 13th, 2005, 5:37pm by Dis » |
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pcbouhid
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #2 on: Dec 13th, 2005, 6:11pm » |
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? ? ? ? 
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| « Last Edit: Dec 13th, 2005, 6:12pm by pcbouhid » |
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SMQ
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #3 on: Dec 13th, 2005, 6:44pm » |
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He said a) x is 823525 
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pcbouhid
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #4 on: Dec 14th, 2005, 4:50am » |
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I wonder if it would be more easy to write 823,525.
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Grimbal
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #5 on: Dec 14th, 2005, 6:20am » |
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It could be if you don't have a compiler...
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JohanC
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #6 on: Dec 14th, 2005, 6:27am » |
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What about
b) 2,705,909 ?
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pcbouhid
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #7 on: Dec 14th, 2005, 7:21am » |
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right, JC.
Talking about compilers: I have no knowledge in programming PC´s, using "C' or "C+", or whatever. As I said once, I´m from the time of Cobol, Assembler, Fortran, etc...
Anybody can direct me how to have in my computer a compiler of the common languages used today? I´m always reading a program in a language that has no meaning to me, but I want to learn something about.
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towr
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #8 on: Dec 14th, 2005, 8:03am » |
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You don't need a compiler per se.
If you have a browser, you can do small programs in javascript
open a new window/tab, and where you usually put the url, put
javascript:document.write("hello world"); document.close();
voila, first program, easy as that.
But if you want to go for a grown-up language, under Linux you standard have gcc/g++, also javac if I'm not mistaken. For windows it's often a bit more work to get a free compiler.
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| « Last Edit: Dec 14th, 2005, 8:04am by towr » |
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JohanC
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #9 on: Dec 14th, 2005, 9:22am » |
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on Dec 14th, 2005, 7:21am, pcbouhid wrote:| Anybody can direct me how to have in my computer a compiler of the common languages used today? I´m always reading a program in a language that has no meaning to me, but I want to learn something about. |
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If your main goal is learning, and solving/checking some of your own mathematical puzzles, you'ld probably also interested in a program such as Pari-GP.
It can be downloaded (together with its manuals) for free at:
http://pari.math.u-bordeaux.fr/
You can either use it as a powerful calculator (including very large numbers, fractions, polyonomes, ...) or as a programming environment.
The syntax is quite similar to C, but more limited (I mean for programming; its much more extensive for mathematical stuff).
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pcbouhid
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #10 on: Dec 14th, 2005, 11:20am » |
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Tks towr and JC !!
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towr
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #11 on: Dec 14th, 2005, 1:50pm » |
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If you like logic, you could also give Prolog a try (e.g. http://www.swi-prolog.org/). It's a logical programming language, and quite nice for some problems (I've used it a few times to solve puzzles on this board as well, so there should be a few examples around)
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| « Last Edit: Dec 14th, 2005, 1:51pm by towr » |
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Icarus
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #12 on: Dec 14th, 2005, 6:12pm » |
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Since we are discussing languages... I have in the last couple of years finally come to grips with object-oriented languages, perforce with Visual Basic. The more I use it, however, the more I yearn for an object-oriented language wherein the objects are natural, the language is powerful, and the syntax was not dreamed up by some spastic with a hatred for all mankind.
I've had thoughts about how a language could be constructed in which all data types are objects, and all operators (including assignment) are methods. Such a language would interpret the expression "x = y" as "x.=(y)". I am curious if such a language has been developed.
So, does anyone know of such a language? Regardless of how far it goes in that direction, what is your favorite object-oriented language? And, most importantly, where can I obtain a free (i'm cheap), but legal, compiler for a Windows version?
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Barukh
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #13 on: Dec 14th, 2005, 11:14pm » |
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Icarus, I was told that Smalltalk is an ultimative object-oriented languange. At least, it has some of the features you wanted. See the Wikipedia article for details and links.
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towr
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #14 on: Dec 15th, 2005, 12:38am » |
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on Dec 14th, 2005, 6:12pm, Icarus wrote:| Such a language would interpret the expression "x = y" as "x.=(y)". I am curious if such a language has been developed. |
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That's generally how C++ does it. Although the simple types (int, bool etc) aren't generally regarded as classes, but you could wrap them up easily enough.
Quote:| what is your favorite object-oriented language? |
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Well, a few years ago I'd have said MOO, but I can't use that easily anymore. It is not in fact that great either, but it was convenient for small puzzles and such.
I suppose C++ is the only OO language I regularly work with at the moment. Although more often that not I forego the OO paradigm, since often not usefull for small programs.
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JohanC
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #15 on: Dec 15th, 2005, 2:52am » |
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Lots of puzzlers seem to be very fond of Haskell, a functional language based on lambda calculus.
Does somebody here have hands-on experience?
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towr
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #16 on: Dec 15th, 2005, 3:53am » |
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on Dec 15th, 2005, 2:52am, JohanC wrote:Lots of puzzlers seem to be very fond of Haskell, a functional language based on lambda calculus.
Does somebody here have hands-on experience? |
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A little bit, but that was a long while ago. There's a chance I'll get better aquainted with it in a little while in the project I'm working on.
LISP (and also Scheme) is an other lambda calculus based language. Not so popular for the main stream though, too many parentheses 
The benefits of functional, and also logical, languages over imperative and object oriented ones is that they often yield shorter and more elegant programs.
But it depends on what sort of problem you're dealing with. If you want to use iteration, it can get messy (because you often have to deal with it recursively).
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rmsgrey
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #17 on: Dec 15th, 2005, 9:18am » |
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For the computing course for my degree, I did some ML - writing code for it feels very natural to a trained mathematician...
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Margit
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #18 on: Dec 16th, 2005, 12:33am » |
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Icarus, You might like to check out Ruby -
www.ruby-lang.org
Quote from features -
"Ruby is a complete, full, pure object oriented language: OOL. This means all data in Ruby is an object, in the sense of Smalltalk: no exceptions. Example: In Ruby, the number 1 is an instance of class Fixnum."
Gcc has several frontend addons which normal Linux distros include and which may or may not be installed on your Linux system. Some of these are :
C++, D, ADA, Java, Pascal, Modula 2, Fortran etc.
For Win users who would like to use open source compilers and whatever else, check out :
www.cygwin.com
This gives you a Linux environment under Win.
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pcbouhid
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Re: A PROBLEM ON REMAINDERS AND QUOTIENTS
« Reply #19 on: Dec 16th, 2005, 8:26am » |
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Can I change the title to "A PROBLEM ON COMPILERS, LINUX AND WINDOWS?"
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