There are three types of questions that are based on LCM with remainders. They are as follows.

**When the remainders are same for all the divisors**

In this case the required number will be the LCM × N + remainder, where N is any natural number

N=1, will give smallest such number

N=2, will give second smallest such number and so on-
**When the remainders are different for different divisors but the respective difference between the divisors and the remainders remain constant.**

In this case the required number will be LCM × N +difference of any (divisor - remainder)

refer to example 2. -
**When neither the divisors are same nor the respective difference between the divisors and the remaniders remain constant.**

In this case solve the question by forming equations and solving them.

**Question 1**What is the least possible number which when divided by 4,5,6 leaves the remainder as 3,4,5 respectively.

59 | 60 |

61 | 119 |

**Question 2**Find the least possible 5 digit number which when divided by 2,4,6,8 it leaves the remainder 1,3,5,7 respectively.

10006 | 10007 |

10008 | None of these |

**Question 3**What is the least possible number which when divided by 10,12,14 leaves the remainder as 2. How many such numbers are there between 5000 and 6000.

1 | 2 |

3 | 4 |

**Question 4**What is the least possible number which when divided by 11 leaves the remainder 3 and when divided by 5 leaves the remainder as 2.

46 | 47 |

48 | 49 |

**Question 5**What will be the least possible number which when divided by 4,5,6 always leaves the remainder 3. Find the following such numbers which satisfy the given condition.

i) Smallest

ii) Second Smallest

iii) Greatest number smaller than 1000

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