You are welcome, I am happy if the table can help. It's not difficult to extend it to - 7 to 7, for instance, so covering the - 7 to 1 or the -1 to 7 book puzzles.

However, do not forget that you could find situations (depending on the software or, in our site, until Patrick clarifies about the sign of the remainder, ... ), when dealing with mod operation and negative numbers, where two different results can be obtained, both correct, and both complying with |r| < |d| (as exposed in the Wikipedia), so there is, let's say, a "duplicity" or "duality" (in the remainder and in the quotient).

Those two solutions are part of the same class of "equivalence". For instance, you can see in the table that -4 mod -5 = - 4 (choosing the "natural" "integer" 0 as the quotient) but if you choose a quotient of 1 then 1 x (- 5) = - 5 and the remainder is: - 4 - (- 5) = - 4 + 5 = 1. In the class of equivalence created by the divisor - 5, the - 4 and the 1 are both representative members, exactly as it is the - 14, for instance, quotient 2 >>> 2 x (- 5) = - 10 >>> remainder is - 14 - (- 10) = - 14 + 10 = - 4. Or the - 9, another representative of the "class", quotient 1 >>> 1 x (- 5) = - 5 >>> remainder: - 9 - (- 5) = - 9 + 5 = - 4, etc. Using different quotients is like using the subtraction process to create representatives of the "class".

Consequently in the table provided each result has another "colleague", "in the shadow", as a possible solution of the mod operation (both absolute values below the absolute value of the divisor). The modulo operation is easily understood with positive numbers while with negative numbers the "field of possibilities" expand to all numbers "to the left" of zero .

Best, clm.

Hi clm,
it' possible to extend the table to 7 (between -5 and 7) ? if it's not too long for you...
Because I'm afraid of making mistakes...

And a table more extended, from -7 to 7 (replacing the .5 to 7 table):
________ - 7_____- 6______- 5_____- 4_____- 3______- 2____- 1______0_____1_____2_____3_____4_____5_____6______7

___- 7____________- 1______- 2______- 3______- 1______- 1______0______N/A_____0_____- 1____- 1_____- 3____- 2_____-1______0
___- 6___- 6_______________- 1______- 2______ 0________0______0______N/A_____0______0_____0_____- 2_____- 1_____0______-6
___- 5___- 5______- 5_______________- 1______- 2______- 1______0______N/A_____0_____- 1____- 2_____- 1_____0_____-5______-5
___- 4___- 4______- 4______- 4_______________- 1_______0______0______N/A_____0______0_____- 1______0____- 4_____-4______-4
___- 3___- 3______- 3______- 3______- 3_______________- 1______0______N/A_____0_____- 1_____0_____- 3____- 3_____-3______-3
___- 2___- 2______- 2______- 2______- 2______- 2_______________0______N/A_____0______0____- 2_____- 2____- 2_____-2______-2
___- 1___- 1______- 1______- 1______- 1_______-1_______-1_____________N/A_____0_____- 1____- 1_____- 1____- 1_____-1______-1
____ 0____0_______0_______0________0_______0________0______0______N/A_____0______0_____0_______0_____0______0______0
____ 1____1_______1_______1________1_______1________1______0______N/A____________1_____1_______1_____1______1______1
____ 2____2_______2_______2________2_______2________0______0______N/A_____0____________2_______2_____2______2______2
____ 3____3_______3_______3________3_______0________1______0______N/A_____0______1_____________3_____3______3______3
____ 4____4_______4_______4________0_______1________0______0______N/A_____0______0_____1_____________4______4______4
____ 5____5_______5_______0________1_______2________1______0______N/A_____0______1_____2_______1____________5______5
____ 6____6_______0_______1________2_______0________0______0______N/A_____0______0_____0_______2_____1_____________6
____ 7____0_______1_______2________3_______1________1______0______N/A_____0______1_____1_______3_____2______1_______

The blank spaces in the main diagonal because obviously we cann't repeat the same number twice in a 2-cell cage.

cool, gracias gracias gracias, clm, that's sweet.