# Using "Outer" with "ChineseRemainder"?

I'm looking to build a $7\times3$ matrix using ChineseRemainder. Trying to use

${0,1,2,3,4,5,6} \pmod 7$

and

${3,15,21} \pmod {30}$

so that the first element of the matrix is the answer to

$0 \pmod 7$

and

$3 \pmod {30}$

or $63$. Thus in ChineseRemainder form,

ChineseRemainder[{0, 3}, {7, 30}]


How would I go about writing this? I figured I'd have to use Outer with ChineseRemainder but can't seem to make it work.

• Perhaps Outer[ChineseRemainder[{##}, {7, 30}] &, {0, 1, 2, 3, 4, 5, 6}, {3, 15, 21}] ? Apr 11, 2016 at 16:54
• @ilian That did it! Added MatrixForm and it spit out what I needed! Could you maybe put this as an answer and include a description of what the ## and & do in coordination with Outer? I'll gladly mark as best answer. Apr 11, 2016 at 17:49

As already mentioned in the question, we can use Outer, for example (with a $3 \times 2$ matrix)

Outer[f, {x1, x2, x3}, {y1, y2}]

{{f[x1, y1], f[x1, y2]}, {f[x2, y1], f[x2, y2]}, {f[x3, y1], f[x3, y2]}}


so we just need a function f for which f[x, y] returns ChineseRemainder[{x, y}, {7, 30}].

That function could be defined simply as

f[x_, y_] := ChineseRemainder[{x, y}, {7, 30}]


and then the desired result is

Outer[f, {0, 1, 2, 3, 4, 5, 6}, {3, 15, 21}]


It is slightly shorter to use a pure (or anonymous) Function where ##, or SlotSequence, denotes the sequence of arguments, e.g.

g[{##}, {m, n}] & [x, y]

(* g[{x, y}, {m, n}] *)


where the head g could be, say ChineseRemainder. Putting this together,

mat = Outer[ChineseRemainder[{##}, {7, 30}] &, {0, 1, 2, 3, 4, 5, 6}, {3, 15, 21}]


and of course mat can be displayed as a matrix by MatrixForm[mat].

Tuples can be a useful alternative to Outer,e.g. the 7, 30 case:

r = Range[0, 6];
c = {3, 15, 21};
res = ChineseRemainder[{#1, #2}, {7, 30}] & @@@ Tuples[{r, c}];
TableForm[Partition[res, 3], TableHeadings -> {r, c}]


• This is great! Very neat, orderly, and well labeled! I'm a little confused tho, on what exactly it does, and at present wouldn't be build something similar on my own. I get r and c (and might change r as follows "n=7; r=Range[0,n-1];" then use "{n,30}" instead of {7,30} so both will be set as I set a value for "n".) For "res", are #1 and #2 referring to the above lines for "r" and "c"? What does "& @@@ Tuples" do? Does "Partition" define the number of columns? I know I could look this up, but there's enough here I don't fully get as a newbie, I'd be a little overwhelmed. Thanks Apr 12, 2016 at 19:14
• @Elem-Teach-w-Bach-n-Math-Ed thank you. Apologies for delay in replying: timezone. Tuples just does what outer does but flattens the list. @@@ is useful. It changes the head of list. ,e.g. f@@@{{a,b},{c,d}}->{f[a,b],f[c,d]}. Just dive in and play. There are many ways to get things done. Sometimes terse looks nice but is inefficient. I learn a lot from MSE. So have fun. :) Apr 12, 2016 at 22:48