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Just tried googling but couldn't find any example, but how $3\equiv 3\bmod{5}$

Googled it

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If you count $1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,\ldots$, what is the third number that comes up? – Michael Greinecker Sep 12 '12 at 10:30
When you divide $3$ by $5$, the remainder is $3$. – Joe Johnson 126 Sep 12 '12 at 10:31
Also, $3 \equiv 3 \mod 5$, because $3-3=0 \equiv 0 \mod 5$. – Andrea Orta Sep 12 '12 at 10:34
If you are asking this question $\implies$ you need to look at the definition of modulo operation. see – Aang Sep 12 '12 at 12:45
up vote 13 down vote accepted

If $a$ and $b$ are positive integers, there exist unique integers $q$, $r$ with $$a = bq + r$$ and $0 \leq r < b$.

This theorem is called the division algorithm, and $a\ \%\ b$ is defined to be this $r$. In your case, $3 = 0\cdot5 + 3$ and $0 \leq 3 < 5$, so the answer is 3.

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$a \equiv b \, (\text{mod} c)$ means $a-b$ is divisible by $c$. [Definition]

Since $3 - 3 = 0$ is divisible by $5$, we have $3 \equiv 3 \, (\text{mod} 5)$.

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hmmm Actually Mod (%) returns the remainder Given two positive numbers, a (the dividend) and b (the divisor), a modulus % is the remainder of the Euclidean division of a by b. For instance, the expression "9 mod 8" would evaluate to 1 because 9divided by 8 leaves a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 leaves a remainder of 0. hope this will help you cheers

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plus if your dividend if less than the divisor, then your remainder will be the dividend :) – Ali Sep 12 '12 at 10:52
It works for negative numbers too, unless you are using a non-standard definition on Mod? – user1729 Sep 12 '12 at 13:02

a modulo b = r Where r is the remainder of the division a by b => a = b*q + r where q is the quotient of the division a/b.

For example: 9 modulo 4 = 1 and the quotient is 2

i.e 9 = (4*2) + 1

In a similar fashion: 3 modulo 5 = 3 and the quotient is 0- =>3=(5*0)+3

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in general if a is less than b then a%b= a

so this implies that 3%5=3 and 6%7=6. another way of viewing this is by reading the statement a%b as "how many object remains if we SUCCESSIVELY take "b" objects from a bag with "a" objects untill we CAN'T TAKE ANYMORE!". Having that in mind 5%2 mean that we successively take 2 objects from a bag containing 5 objects. 1st we will take the 2 objects and leaving behind 3 objects, then we take another 2 object leaving behing 1 object in the bag, since we can't take 2 objects any more, SINCE there is only one remaining hence 5%2=1. applying the same analogy and solving for 3%5 we initially have 3 object in a bag and we want to successively take 5 objects. but we cant since the are insufficient objects in the bag so we can't take any objects and hence 3 objects will remain, this implies that 3%5=3.

hope this helps.

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Check out the following link to help format questions/answers. – Vincent Oct 21 '14 at 16:13
The only answer that actually made sense. When people ask these kinds of questions, it's because of a lack of training in mathematics and the last thing they need are mathematicians trying to explain things with more math. Thanks for breaking it down to simple terms! :) – kirk.burleson Jul 19 '15 at 16:21

by definition, the remainder of a division is the fractional part of a division. if you have 3/5 (3 divided by 5), you have the fractional part represented as 3/5. since a remainder is the dividend or numerator of a fraction, you just take 3 as your remainder (the divisor or denominator is not mentioned in the remainder).
therefore, 3 % 5 = 3.

just like 5 % 3 = dividing 5 by 3, you get 1 and 2/5. to get the remainder part, you know that 2 is the numerator so 2 is the remainder (the denominator, 5, is not mentioned in the remainder).

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