Understanding The Modulus Operator

Question

I understand the Modulus operator in terms of the following expression   7   5   This would return 2 due to the fact that 5 goes into 7 once and then gives the 2 that is left over  however my confusion comes when you reverse this statement to read   5   7   This gives me the value of 5 which confuses me slightly  Although the whole of 7 doesn t go into 5  part of it does so why is there either no remainder or a remainder of positive or negative 2    If it is calculating the value of 5 based on the fact that 7 doesn t go into 5 at all why is the remainder then not 7 instead of 5   I feel like there is something I m missing here in my understanding of the modulus operator

User · Answer

lets put it in this way   actually Modulus operator does the same division but it does not care about the answer   it DOES CARE ABOUT reminder for example if you divide 7 to 5    so   lets me take you through a simple example   think 5 is a block  then for example we going to have 3 blocks in 15  WITH Nothing Left    but when that loginc comes to this kinda numbers  1 3 5 7 9 11        here is where the Modulus comes out   so take that logic that i said before and apply it for 7   so the answer gonna be that we have 1 block of 5 in 7    with 2 reminds in our hand  that is the modulus     but you were asking about 5   7   right    so take the logic that i said   how many 7 blocks do we have in 5      0  so the modulus returns 0     that s it

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As you say  the   sign is used to take the modulus  division remainder    In w3schools  JavaScript Arithmetic page we can read in the Remainder section what I think to be a great explanation     In arithmetic  the division of two integers produces a quotient and a   remainder       In mathematics  the result of a modulo operation is the   remainder of an arithmetic division    So  in your specific case  when you try to divide 7 bananas into a group of 5 bananas  you re able to create 1 group of 5  quotient  and you ll be left with 2 bananas  remainder    If 5 bananas into a group of 7  you won t be able to and so you re left with again the 5 bananas  remainder

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This explanation is only for positive numbers since it depends on the language otherwise   Definition  The Modulus is the remainder of the euclidean division of one number by another    is called the modulo operation   For instance  9 divided by 4 equals 2 but it remains 1  Here  9   4   2 and 9   4   1      In your example  5 divided by 7 gives 0 but it remains 5  5   7    5    Calculation  The modulo operation can be calculated using this equation   a   b   a - floor a   b    b    floor a   b  represents the number of times you can divide a by b floor a   b    b is the amount that was successfully shared entirely The total  a  minus what was shared equals the remainder of the division   Applied to the last example  this gives   5   7   5 - floor 5   7    7   5   Modular Arithmetic  That said  your intuition was that it could be -2 and not 5  Actually  in modular arithmetic  -2   5  mod 7  because it exists k in Z such that 7k - 2   5   You may not have learned modular arithmetic  but you have probably used angles and know that -90   is the same as 270   because it is modulo 360  It s similar  it wraps  So take a circle  and say that it s perimeter is 7  Then you read where is 5  And if you try with 10  it should be at 3 because 10   7 is 3

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A novel way to find out the remainder is given below   Statement   Remainder is always constant   ex   26 divided by 7 gives R   5    This can be found out easily by finding the number that completely divides 26 which is closer to the  divisor and taking the difference of the both   13 is the next number after 7 that completely divides 26 because after 7 comes 8  9  10  11  12 where none of them divides 26 completely and give remainder 0   So 13 is the closest number to 7 which divides to give remainder 0   Now take the difference  13   7    5 which is the temainder   Note  for this to work divisor should be reduced to its simplest form ex  if 14 is the divisor  7 has to be chosen to find the closest number dividing the dividend

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As others have pointed out modulus is based on remainder system    I think an easier way to think about modulus is what remains after a dividend  number to be divided  has been fully divided by a divisor  So if we think about 5 7  when you divide 5 by 7  7 can go into 5 only 0 times and when you subtract 0  7 0  from 5  just like we learnt back in elementary school   then the remainder would be 5   the mod   See the illustration below      0          7  5       -0        5   With the same logic  -5 mod 7 will be -5   only 0 7s can go in -5 and -5-0 7   -5   With the same token -5 mod -7 will also be -5  A few more interesting cases   5 mod  -3    2  i e  5 -  -3 -1    -5  mod  -3    -2 i e  -5 -  -3 1    -5 3

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Two Steps Solution  Some of the answers here are complicated for me to understand  I will try to add one more answer in an attempt to simplify the way how to look at this   Short Answer  Example 1   7   5   2 Each person should get one pizza slice   Divide 7 slices on 5 people and every one of the 5 people will get one pizza slice and we will end up with 2 slices  remaining   7   5 equals 2 is because 7 is larger than 5   Example 2   5   7   5 Each person should get one pizza slice  It gives 5 because 5 is less than 7  So by definition  you cannot divide whole 5items on 7 people  So the division doesn t take place at all and you end up with the same amount you started with which is 5   Programmatic Answer  The process is basically to ask two questions  Example A   7   5   Q 1  What number to multiply 5 in order to get 7  Two Conditions  Multiplier starts from  0   Output result should not exceed  7     Let s try  Multiplier is zero 0 so  0 x 5   0 Still  we are short so we add one   1  to multiplier  1 so  1 x 5   5 We did not get 7 yet  so we add one   1   2 so  2 x 5   10 Now we exceeded 7  So 2 is not the correct multiplier  Let s go back one step  where we used 1  and hold in mind the result which is5  Number 5 is the key here   Q 2  How much do we need to add to the 5  the number we just got from step 1  to get 7  We deduct the two numbers  7-5   2  So the answer for   7   5 is 2   Example B   5   7  1- What number we use to multiply 7 in order to get 5  Two Conditions  Multiplier starts from  0   Output result and should not exceed  5     Let s try  0 so  0 x 7   0 We did not get 5 yet  let s try a higher number  1 so  1 x 7   7 Oh no  we exceeded 5  let s get back to the previous step where we used 0 and got the result 0  2- How much we need to add to 0  the number we just got from step 1  in order to reach the value of the number on the left 5  It s clear that the number is 5  5-0   5    5   7   5  Hope that helps

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It s just about the remainders  Let me show you how  10   5 0 9   5 4  because the remainder of 9 when divided by 5 is 4  8   5 3 7   5 2 6   5 1  5   5 0  because it is fully divisible by 5    Now we should remember one thing  mod means remainder so  4   5 4   but why 4  because 5 X 0   0 so 0 is the nearest multiple which is less than 4  hence 4-0 4

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Modulus operator gives you the result in  reduced residue system   For example for mod 5 there are 5 integers counted  0 1 2 3 4  In fact 19 12 5 -2 -9  mod 7   The main difference that the answer is given by programming languages by  reduced residue system

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modulus is remainders system    So 7   5   2    5   7   5  3   7   3   2   7   2  1   7   1  When used inside a function to determine the array index  Is it safe programming   That is a different question  I guess

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Step 1   5 7   0 71  Step 2   Take the left side of the decimal   so we take 0 from 0 71 and multiply by 7            0 7   0   Step     5-0   5   Therefore   5 7  5

[modulus] Understanding The Modulus Operator %

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