Why does 2 mod 4 2

Question

I m embarrassed to ask such a simple question  My term does not start for two more weeks so I can t ask a professor  and the suspense would kill me   Why does 2 mod 4   2

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Someone contacted me and asked me to explain in more details my answer in the comment of the question. So here is what I replied to that person in case it can help anyone else:

The modulo operation gives you the remainder of the euclidian disivion (which only works with integer, not real numbers). If you have A such that A = B * C + D (with D < B), then the quotient of the euclidian division of A by B is C, and the remainder is D. If you divide 2 by 4, the quotient is 0 and the remainder is 2.

Suppose you have A objects (that you can't cut). And you want to distribute the same amount of those objects to B people. As long as you have more than B objects, you give each of them 1, and repeat. When you have less than B objects left you stop and keep the remaining objects. The number of time you have repeated the operation, let's call that number C, is the quotient. The number of objects you keep at the end, let's call it D, is the remainder.

If you have 2 objects and 4 people. You already have less than 4 objects. So each person get 0 objects, and you keep 2.

That's why 2 modulo 4 is 2.

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Modulo  mod     is the Remainder operator    2 2   0  2 2   1 remainder 0  1 2   1  1 2   0 remainder 1  4 2   0  4 2   2 remainder 0  5 2   1  5 2   2 remainder 1

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mod means the reaminder when divided by   So 2 divided by 4 is 0 with 2 remaining   Therefore 2 mod 4 is 2

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MOD is remainder operator  That is why 2 mod 4 gives 2 as remainder  4 0 0 and then 2-0 2  To make it more clear try to do same with 6 mod 4 or 8 mod 3

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The modulo operator evaluates to the remainder of the division of the two integer operands  Here are a few examples   23   10 evaluates to 3  because 23 10 is 2 with a remainder of 3  50   50 evaluates to 0  50 50 is 1 with a remainder of 0  9   100 evaluates to 9  9 100 is 0 with a remainder of 9

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I was confused about this  too  only a few minutes ago  Then I did the division long-hand on a piece of paper and it made sense    4 goes into 2 zero times   4 times 0 is 0   You put that zero under the 2 and subtract which leaves 2     That s as far as the computer is going to take this problem  The computer stops there and returns the 2  which makes sense since that s what      mod  is asking for    We ve been trained to put in the decimal and keep going which is why this can be counterintuitive at first

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This could be a good time to mention the modr   function  It returns both the whole and the remainder parts of a division   print   n 17    3    17  3     Does the same thing as int 17 3    print   17    3    17 3     Modulo division gives the remainder    whole  remain   divmod 17 3  print   divmod 17 3  returns - gt   divmod 17 3  end     print   because 3 goes into 17   whole  times with a remainder of  remain end    n n

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Modulo is the remainder  not division   2   4   0R2 2   4   2   The sign   is often used for the modulo operator  in lieu of the word mod   For x   4  you get the following table  for 1-10    x x 4 ------  1  1  2  2  3  3  4  0  5  1  6  2  7  3  8  0  9  1 10  2

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When you divide 2 by 4  you get 0 with 2 left over or remaining  Modulo is just the remainder after dividing the number

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Mod operation works with reminder   This is called modular arithmetic    a  b mod m    then m  a-b   a-b km   a b km  So  2 2 0 4

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This is Euclid Algorithm   e g  a mod b   k   b   c    a mod b   c  where k is an integer and c is the answer  4 mod 2    2   2   0    4 mod 2   0  27 mod 5   5   5   2    27 mod 5   2  so your answer is  2 mod 4   0   4   2    2 mod 4   2

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Mod just means you take the remainder after performing the division  Since 4 goes into 2 zero times  you end up with a remainder of 2

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Much easier if u use bananas and a group of people    Say you have 1 banana and group of 6 people  this you would express  1 mod 6   1   6   1 modulo 6     You need 6 bananas for each person in group to be well fed and happy     So if you then have 1 banana and need to share it with 6 people  but you can only share if you have 1 banana for each group member  that is 6 persons  then you will have 1 banana  remainder  not shared on anyone in group   the same goes for 2 bananas  Then you will have 2 banana as remainder  nothing is shared     But when you get 6 bananas  then you should be happy  because then there is 1 banana for each member in group of 6 people  and the remainder is 0 or no bananas left when you shared all 6 bananas on 6 people     Now  for 7 bananas and 6 people in group  you then will have 7 mod 6   1  this because you gave 6 people 1 banana each  and 1 banana is the remainder    For 12 mod 6 or 12 bananas shared on 6 people  each one will have two bananas  and the remainder is then 0

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For a visual way to think about it  picture a clock face that  in your particular example  only goes to 4 instead of 12   If you start at 4 on the clock  which is like starting at zero  and go around it clockwise for 2  hours   you land on 2  just like going around it clockwise for 6  hours  would also land you on 2  6 mod 4    2 just like 2 mod 4    2

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I think you are getting confused over how the modulo equation is read   When we write a division equation such as 2 4 we are dividing 2 by 4   When a modulo equation is wrote such as 2   4 we are dividing 2 by 4  think 2 over 4  and returning the remainder

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2   4   0 with a remainder of 2

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To answer a modulo x   y  you ask two questions   A- How many times y goes in x without remainder   For 2 4 that s 0   B- How much do you need to add to get from that back to x   To get from 0 back to 2 you ll need 2-0  i e  2   These can be summed up in one question like so  How much will you need to add to the integer-ish result of the division of x by y  to get back at x   By integer-ish it is meant only whole numbers and not fractions whatsoever are of interest   A fractional division remainder  e g   283849  is not of interest in modulo because modulo only deals with integer numbers

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For    2 mod 4  We can use this little formula I came up with after thinking a bit  maybe it s already defined somewhere I don t know but works for me  and its really useful   A mod B   C where C is the answer  K   B - A    C  where K is how many times B fits in A  2 mod 4 would be   0   4 - 2    C   C    -2    gt  2  Hope it works for you

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Modulo is the remainder  expressed as an integer  of a mathematical division expression     So  lets say you have a pixel on a screen at position 90 where the screen is 100 pixels wide and add 20  it will wrap around to position 10   Why   because 90   20   110 therefore 110   100   10   For me to understand it I consider the modulo is the integer representation of fractional number   Furthermore if you do the expression backwards and process the remainder as a fractional number and then added to the divisor it will give you your original answer   Examples       100  A  ---     14 mod 2      7      123  B  ---     8 mod 3      15       3  C  ---     0 mod 3      4   Reversed engineered to                           2      14 7     2       98    2     100   A  14 mod 2     14   ---     -----   ---     ---   ---    ---                         7        7      7       7     7      7                        3      8 15     3      120    3      123   B  8 mod 3     8   ---     -----   ---     ---   ---     ---                       15       15     15      15    15      15                        3       3   B  0 mod 3     0   ---     ---                       4       4

[modulo] Why does 2 mod 4 = 2?

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