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Largest number that leaves same remainder while dividing 5958, 5430 and 5814 ?

$$5958 \equiv 5430 \equiv 5814 \pmod x$$ $$3\times 17\times 19 \equiv 5\times 181\equiv 3\times 331\pmod x$$ $$969 \equiv 905\equiv 993\pmod x$$

After a bit of playing with the calculator, I think the answer is $48$ but I don't know how to prove it.

Sorry if the answer is too obvious, I am still trying to wrap my head modular arithmetic and not very successful yet.It would be great help if anybody would give me some hints on how to proceed ahead. Thanks. $\ddot \smile$

Martin Sleziak
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  • The first congruence corresponds to the title. What do you mean with the other two congruences? Are you solving three problems at once here? – user236182 Aug 25 '16 at 23:13
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    I see you divided all the terms by 6. That's .. not really okay. 12 = 22 mod 10 but 6 != 11 mod 10. – fleablood Aug 26 '16 at 00:03

4 Answers4

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Okay, first we set it up even though we have no idea where we are going.

$5958 \equiv n \mod x$

$5430 \equiv n \mod x$

$5814 \equiv n \mod x$.

Then we noodle a bit to make it smaller and more pallatable.

$5958 - 5814 = 144 \equiv 0 \mod x$

$5814 - 5430 = 384 \equiv 0 \mod x$

and

$5958 - 5430 = 528 \equiv 0 \mod x$

Well, now it's very clear that $x$ is a common divisor and all we have to do is find the $x = \gcd(144,384, 528)$

$x = \gcd(144 = 2^43^2, 384 = 2^7*3, 528 = 2^4*3*11) = 2^4*3 = 48$.

And we're done.

fleablood
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    we could noodle a bit further to get 528 - 384 = 144 = 0 mod x and so on and accidentally stumble onto the Euclidean Algorithm completely inadvertently. – fleablood Aug 25 '16 at 23:35
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    ... then we wonder briefly "well, what was the remainder anyway" and then decide "screw it, I don't care". – fleablood Aug 25 '16 at 23:38
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    ... then we decide the heck with it and calculate that the remainder was $n=6$. – fleablood Aug 25 '16 at 23:39
  • Thanks for clearing my doubts through this answer and the comment below the question. :) –  Aug 26 '16 at 00:14
  • Can you recommend a good book on modular arithmetic with difficult exercises, if you know any? my course is kind of basic and i am getting trouble doing tricky problems. Thanks. :) –  Aug 26 '16 at 08:57
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    Don't really have any recomendations. Any number theory book covers it. Ivan Niven's book was the one I learned from but any will do. – fleablood Aug 26 '16 at 18:03
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Since $5958 \equiv 5430 \mod{x}$, then $x | 5958 -5430 = 528$. Likewise $x | 5958 - 5814 = 144$ and $x | 5814 - 5430 = 384$. So $x$ is a common divisor of $528,$ $144,$ and $384$. So you want $x=\gcd(528, 144, 384) = 48$.

B. Goddard
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Let the number leaving the same remainder be $a$ and the remainder be $b$.

So, $\gcd(5958-a,5430-a,5814-a)=b$

Note that

$(1)$ $$\gcd(x,y,z)=\gcd{\gcd(x,y)\gcd(y,z)}$$ $(2)$ $$\gcd(x,y) \mid x-y$$

Roby5
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  • thanks for the hints, took me a while but got them at last $\ddot \smile$ –  Aug 26 '16 at 00:18
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The only "small" solutions (below 10000000) seem to be ${1,2,3,4,6,8,12,16,24,48}$. Of course, you can add $lcm({5958, 5430, 5814}) = 1741612770 $ to a solution and get another, bigger one.

The set above was found by letting do the computer what a computer can do best: Computing stuff. Here is the Haskell Code I used:

[x | x <- [1..870806385 + 49], 5958 `mod` x == 5430 `mod` x, 5430 `mod` x == 5814 `mod` x]

Kaligule
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    You can't add $1741612770$ to any of these, as it would then leave $5958$ etc. as remainders. – Ross Millikan Aug 25 '16 at 23:32
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    there can't be any solutions greater than 5430 so no point in pointing out that these are only the small ones. If n > 5430 then the remainder dividing into 5430 is 5430. – fleablood Aug 25 '16 at 23:32
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    I think you misunderstood the question or i did not frame it well, either way thanks for putting the effort. –  Aug 26 '16 at 00:15