Least number which when divided by 35,45,55 and leaves remainder 18,28,38; is?

Problem 1: Least number which when divided by 35,45,55 and leaves remainder 18,28,38; is?
Solution: i) In this case we will evaluate l.c.m.
               ii) Here the difference between every divisor and remainder is same i.e. 17.
                  Therefore, required number = l.c.m. of (35,45,55)-17 = (3465-17)= 3448.

Problem 2: Least number which when divided by 5,6,7,8 and leaves remainder 3, but when divided by 9, leaves no remainder?
Solution: l.c.m. of 5,6,7,8 = 840
                 Required number = 840 k + 3
                 Least value of k for which (840 k + 3) is divided by 9 is 2
Therefore, required number = 840*2 + 3
                                            = 1683

Problem 3: Greater number of 4 digits which is divisible by each one of 12,18,21 and 28 is?
Solution: l.c.m. of 12,18,21,28 = 254
               Therefore, required number must be divisible by 254.
               Greatest four digit number = 9999
               On dividing 9999 by 252, remainder = 171
               Therefore, 9999-171 = 9828.