All such numbers?
Logic
Level
4
Find the sum of all positive integers that are equal to 700 times the sum of their digits.
The answer is 21000.
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1 solution
Clearly, it is impossible for any integers less than 1000 to satisfy the conditions stated. Also, one can deduce that any integer satisfying the above conditions will end with 2 zeroes. Furthermore, integers with 6 digits will have a minimum digit sum of 143 in order to for 700 times the digit sum to be larger than 100000. However, the maximum possible digit sum is 4*9=36. Therefore, for integers larger than 100000, they will always exceed 700 times their digit sum.
Now, we simply need to consider 4 and 5 digit numbers. For 5 digit numbers, let the first 3 digits be a, b, c respectively. Then, we get 10000a+1000b+100c=700a+700b+700c. Simplifying, we get 93a+3b=6c, which has no solutions. Therefore, we are left with 4 digit numbers which can be obtained simply by checking all four digit multiples of 700.
Thus, the answer is 2100+4200+6300+8400=21000.