The number of natural numbers less than 10000 which are divisible by 5 and that no digit is repeated in the same number, is

Natural numbers less than 10000 which are devisible by 5 , will be the numbers upto 4 digits which ends with zero or 5. Number of all natural numbers upto 4 digits which ends with zero when no digits are repeated $$ =(9)(8)(7)(1)+(9)(8)(1)+(9)(1)=585 $$ (Note that zero will not be counted as $0 \notin \mathbb{N}$ ) Number of all Natural number 5 upto 4 digits which ends with 5 when no digits are repeated. $$ =(8)(8)(7)(1)+(8)(8)(1)+(8)(1)+(1)=521 $$ Therefore, required number of Natural number $$ =585+521=1106 $$