Quick Review: Remainder Theory

The basic remainder formula is:
Dividend = Divisor* Quotient + Remainder
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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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Important Points
  • The sum of consecutive five whole numbers is always divisible by 5.
  • The square of any odd number when divided by 8 will leave 1 as the remainder.
  • The product of any three consecutive natural numbers is divisible by 8.
  • The unit digit of the product of any nine consecutive numbers is always zero.
  • For any natural number n, 10n-7 is divisible by 3.
  • Any three-digit number having all the digits same will always be divisible by 37.
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