Dividend = Divisor* Quotient + Remainder

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.

- (a
^{n }+ b^{n}) is divisible by (a + b), when n is odd. - (a
^{n }- b^{n}) is divisible by (a + b), when n is even. - (a
^{n}- b^{n}) is always divisible by (a - b), for every n.

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 is the property of remainders, due to which they start repeating themselves after a certain point.

Number | Cyclicity |
---|---|

1 | 1 |

2 | 4 |

3 | 4 |

4 | 2 |

5 | 1 |

6 | 1 |

7 | 4 |

8 | 4 |

9 | 2 |

10 | 1 |

Euler’s Remainder theorem states that, for co-prime numbers M and N, Remainder [M^{E(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.

- 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, 10
^{n}-7 is divisible by 3. - Any three-digit number having all the digits same will always be divisible by 37.