Re: Modular Arithmetic Problem?
- To: mathgroup at smc.vnet.net
- Subject: [mg39002] Re: [mg38975] Modular Arithmetic Problem?
- From: Dr Bob <drbob at bigfoot.com>
- Date: Fri, 24 Jan 2003 05:04:52 -0500 (EST)
- References: <200301231304.IAA11717@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Is this what you did (essentially)?
Cases[(Sqrt[FromDigits[#1]] & ) /@
Flatten[
Outer[List, Range[9], {0}, Range[0, 9], Range[0, 9], {1}],
4], _Integer]
%*%
{101, 201, 301}
{10201, 40401, 90601}
I haven't thought of a much smarter method, as yet, but here's a far less
exhaustive search:
nSquare = Expand[FromDigits[{a, b, c}]^2]
nSquare /. List /@ Thread[c -> {1, 9}]
Flatten[% /. List /@ Thread[a -> {1, 2, 3}]]
At this point, it's clear that b=0, c=1, a=1, 2, or 3 gives three valid
answers and that b can't be anything but 0 when a = 3, which narrows the
field considerably.
nSquare /. List /@ Thread[c -> {1, 9}]
Flatten[% /. List /@ Thread[a -> {1, 2}]]
Flatten[% /. List /@ Thread[b -> Range[9]]]
Visual inspection shows no solutions among these, which can be verified as
follows.
Cases[%, q_ /; 10^4 < q < 10^5]
Cases[%, q_ /; Mod[Floor[q/10^3], 10] == 0]
Alternatively, the following isn't too exhaustive, either:
nSquare = Expand[FromDigits[{a, b, c}]^2]
nSquare /. List /@ Thread[c -> {1, 9}]
Flatten[% /. List /@ Thread[a -> {1, 2, 3}]]
Flatten[% /. List /@ Thread[b -> Range[0, 9]]]
Cases[%, q_ /; 10^4 < q < 10^5]
Cases[%, q_ /; Mod[Floor[q/10^3], 10] == 0]
Bobby
On Thu, 23 Jan 2003 08:04:38 -0500 (EST), David Park <djmp at earthlink.net>
wrote:
> Dear MathGroup,
>
> Steven Shippee asked me about methods of solving the following problem.
>
> Find the numbers n such that n^2 is a five digit number with 0 in the
> second
> digit and 1 in the last digit. (i.e., x0xx1 where x is a digit 0..9).
>
> We know how to solve the problem by testing a list of all possible
> candidate
> numbers. But is there a method that uses Solve and modular arithimetic or
> some other clever method? If one looks at the answers one would think
> there
> must be.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/
>
>
>
--
majort at cox-internet.com
Bobby R. Treat
- References:
- Modular Arithmetic Problem?
- From: "David Park" <djmp@earthlink.net>
- Modular Arithmetic Problem?