# Finding Remainders

We can use the following rules to find the remainder when a number is divided by another.

**Rules for Finding Remainders**

**Rule 1 : x is divided by y** When we have to divide a number by another number, then we
can write x in terms of y, i.e. (y×a)+b. This means that the remainder is b.

**e.g.**When 66 is divided by 10, we can write 66 as 10×6+6, thus the remainder is 6.

**Rule 2 : x ± y is divided by z** When we have to divide the sum or difference of two
number by another number, then we write both the numbers as the product of the number with which we have to
divide.

**e.g.**When we have to divide 45+23 by 8, then we write it like

8×5+5 + 8×2+7=8(5+2)+ 12

Since 8(5+2) is divisible by 8 thus we will divide 12 by 8, and we know the remainder will be 4.

Similarily we can solve for 45-23 divided by 8.

**Rule 3 : x × y is divided by z** When we have to divide the product of two numbers by
another number, then the remainder will be equal to the remainder of each of the number divided separately.

**e.g.**When 61×109 is divided by 7, the remainder of is
5 and remainder of is 4, thus we can say the remainder will be

**Rule 4 : x ^{y} is divided by z** When we have to divide a number to some power
with another number, then keep dividing the number with the divisor till the remainder is not less than the
divisor.

**e.g.** When 7^{47} is divided by 5,
will give the remainder as 2, thus 7^{47} divided by 5 will give remainder as

i.e 47 times 2.

Now, 2^{47}
= 2^{3}×2^{411}=

gives the remainder as 3 and gives the remainder as 1.

∴ gives the remainder as 3.

**Rule of Negative Remainder** When a number x divided by y, then we can say that the
remainder is -a if y×k=x+a, where k can be any natural number and a is smaller than y. This in turn means that
the remainder is a times smaller than y.

**e.g** When 8 is divided by 5, we can that remainder is -2. It means 2 less than the divisor *i.e* 3

**e.g** When 120 is divided by 11, we can that remainder is -1. It means 1 less than the divisor *i.e* 10

**Question 1**What will be remainder when 3

^{97!}is divided by 80

2 | 1 |

3 | 4 |

**Question 2**What will be remainder when 1234 × 5678 is divided by 11

1 | 2 |

4 | 6 |

**Question 3**What will be remainder when 75

^{7575}is divided by 37

4 | 3 |

2 | 1 |

**Question 4**What will be remainder when 1!+2!+3!+4!..100! is divided by 7

4 | 5 |

6 | 7 |

**Question 5**What will be remainder when 7

^{486}is divided by 4

0 | 1 |

2 | 3 |

**Question 6**What will be remainder when 1234+5678 is divided by 5

1 | 2 |

3 | 4 |

**Question 7**What will be remainder when 1719 × 1721 × 1723 × 1725 × 1727 is divided by 18

3 | 6 |

8 | 9 |