Question

81. The greatest number of four digits which when divided by 3, 5, 7, 9 leaves the remainders 1, 3, 5, 7 respectively is

Answer 1

Given : Greatest number of 4-digit which when divided by 3 , 5 , 7 , 9 and leaves remainder 1 , 3 , 5 , 7 respectively,

We find LCM of 3 , 5 , 7 and 9 :

Here , 3 = 1 ×× 3 ,

5 = 1 ×× 5 ,

7 = 1 ×× 7

And

9 = 3 ×× 3

So ,

LCM ( 3 , 5 , 7 and 9 ) = 3 ×× 3 ×× 5 ×× 7 = 315

We know largest four digit number = 9999 .

And

9999315 = 31 2343159999315 = 31 234315 So ,

315 ×× 31 = 9765 , That is largest four digit number that is evenly divisible by 3 , 5 , 7 and 9

To get remainders 1 , 3 , 5 , 7 when divided by 3 , 5 , 7 , 9 respectively we find difference of these numbers , As :

3 - 1 = 2 ,

5 - 3 = 2 ,

7 - 5 = 2 ,

9 - 7 = 2

Therefore,

Greatest number of 4-digit which when divided by 3 , 5 , 7 , 9 and leaves remainder 1 , 3 , 5 , 7 respectively = 9765 - 2 = 9763 ( Ans )