SOLUTION: What is the answer for this question? A string of whole numbers is written as follows: 123456789(10)(11)(12)(13)(14)....... find the string with lowest value of such whole numbe

Question 927570: What is the answer for this question?
A string of whole numbers is written as follows: 123456789(10)(11)(12)(13)(14)....... find the string with lowest value of such whole numbers which is divisible by 72.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Assuming we do not count numbers such as 123456789101 or 123...101112131:

Note that a positive integer is congruent to the sum of its digits (mod 9), so the sum of the digits of 1234567891011...n is congruent to 1+2+3+...+n (mod 9). We want 1+2+3+...+n to be divisible by 9 since the string is divisible by 72. Since 1+2+3+...+n = n(n+1)/2, either n or n+1 is divisible by 9.

Also, we want the string to be divisible by 72, so the last three digits must form a multiple of 8. Checking n = 8, 9, 17, 18, 26, 27, 35, 36, we see that n = 36 works because the last three digits are 536 which is a multiple of 8. The shortest string is 12345...343536.
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