Question

How many positive integers less than 1000 have the property that the sum of the
digits of each such number is divisible by 7 and the number itself is divisible by 3?

Answer 1

The number should be divible by 3 so the sum of the digits should be divisible by 3....also given that the sum of digits should be divisible by 7......... therefore the sum of number should b LCM of 3 and 7 i.e. 21........there are no 1-digit or 2-digit number whose sum is 21.......so we take numbers such that 99<n<1000........the possible numbers are: 993 984 975 966 876 858 777

Now 993 966 and 858 can be written in 3!/2!= 3 forms
984 975 876 can be written in 3!= 6 forms
And 777 can be written in one form

So the total numbers are: 3*3 + 6*3 + 1 = 28