Answer to Question #95449 in Combinatorics | Number Theory for Aravindan

Question

Find the greatest number of four digits when divided by 3,5,7,9 leaves remainder 1,3,5,7 respectively

Answer 1

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

9999/315 = 31*234/315 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