find the smallest possible number which is...?

Question

find the smallest possible number which is divisble by 3,5,7 and also its sum of digits of the number are divisible by 3,5,7.

Answer 1

2,899,999,999,995

Method: Since the sum of the digits is divisible by 3, 5, and 7, it must also be divisible by their least common multiple, which is 105. This necessitates that the number have at least 12 digits, since the highest sum of digits an 11-digit number can have is 99. Further, since the number itself must be divisible by 5, the last digit must be a 5 or a 0. However, if the last digit were 5 or less, the largest sum of digits a 13-digit number may have is 9*11+5 or 104, which is short of the required 105. Therefore it must have at least 13 digits. If the first digit is a 1, then for the sum of all the digits to be 105 and the last digit to be no greater than 5, the only possible number is 1,999,999,999,995, which fails to be divisible by 7. Therefore, the first digit must be at least two. If the first digit is two, then since the last digit may be at most 5, this means that at most one of the middle digits may be something other than a 9, and that digit must be an 8 (otherwise, the digits will still sum to less than 105). The smallest 13 digits number that starts with a 2, ends with a 5, and contains 10 9s and 1 8 is 2,899,999,999,995, which happens to fulfill all the requirements. Therefore, 2,899,999,999,995 is the smallest number which is divisible by 3, 5, and 7, and for which the sum of the digits is also divisible by 3, 5, and 7. Q.E.D.

Answer 2

m.c.m. (3,5,7) = 3x5x7 = 105. You need to find a number whose digits add up to 105. It's going to be quite an impressive number by the look of things.

Answer 3

140

Answer 4

Sorry. Just dont tell me you need the answer very badly

Answer 5

You got my head hurtin'.