An integer n is such that [ n 9 ] is a three digit number with equal digits, and [ n-172 4 ] is a 4 digit number with the digits 2,0,2,4 in some order. What is the remainder when n is divided by 100 ?

Step-by-step solution

aligned & Let [ n-172 4 ]=k, k I & array l n-172 4 [k, k+1) n-172 [4 k, 4 k+4) n [4 k+172,4 k+176) n 9 [ 4 k+172 9 , 4 k+176 9 ) array & Now, [ n 9 ] 111,222,333, , 999 & 4 k+176 9 > a a a > 4 k+172 9 , for a 1, , 9 & For a=9 & k (2203.75,2204.75) & k=2204 aligned [ n-172 4 ]=2204 ......(i) And [ n 9 ]=999 ......(ii) From (i): aligned & n-172 4 [2204,2205) & n [8988,8992) aligned From (ii): aligned & n 9 [999,1000) & n [8991,9000) & n=8991 & 91 aligned