If remainder = 0, then it the number is perfectly divisible by divisor and divisor is a factor of the number e.g. when 8 divides 40, the remainder is 0, it can be said that 8 is a factor of 40.
Formulas Based Concepts for Remainder:
(an + bn) is divisible by (a + b), when n is odd.
(an - bn) is divisible by (a + b), when n is even.
(an - bn) is always divisible by (a - b), for every n.
Concept of Negative Remainder:
By definition, remainder cannot be negative. But in certain cases, you can assume that for your convenience. But a negative remainder in real sense means that you need to add the divisor in the negative remainder to find the real remainder.
Cyclicity in Remainders:
Cyclicity is the property of remainders, due to which they start repeating themselves after a certain point.
Cyclicity Table:
Number
Cyclicity
1
1
2
4
3
4
4
2
5
1
6
1
7
4
8
4
9
2
10
1
Role of Euler’s Number in Remainders:
Euler’s Remainder theorem states that, for co-prime numbers M and N, Remainder [ME(N) / N] = 1, i.e. number M raised to Euler number of N will leave a remainder 1 when divided by N. Always check whether the numbers are co-primes are not as Euler’s theorem is applicable only for co-prime numbers.
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