Question

How many positive integers less than 1000 have the property that each digit of the number is divisible by 7 and the number is divisible by 3
Find the total number of integer

Answer 1

The logic for this question is as follows,
We want the numbers to be divisible by 3 which means the sum of the digits should be a multiple of 3. ------- (1)
Now, the max sum for a 3 digit number's digits is 27 (999)
Now, for the sum of digits to be divisible by 7 they should be multiple of 7 ------(2)
Combining 1 and 2, the sum of digits should be multiple of 21 (LCM of 3 and 7)
between 1 to 27, 21 is the only possibility.
Now we just have to find combinations for the sum of three digit numbers whose digits' sum is 21.

Their are total 22 positive integers which are less than 1000 and have property that the sum of digits of each no. is divisible by 7 and the no. itself divisible by 3..
399,489,498,579,588,597,669,678,687,696,786,795,849,858,867,876,885,939,948,957,966 and 975..