Quandaries and Queries


Jack's social security number contains each of the nonzero digits exactly once. By examining the digits from left to right, he also found that 1 divides the first digit evenly, 2 divides the sum of the first two digirs evenly, 3 divides the sum of the first three digits evenly, and so on, until 9 divides the sum of all the digits evenly. What is Jack's social security number?



Jack's social security number looks like


where each letter is a digit between 1 and 9, and no digit is repeated.

The sum of the digits is

a+b+c+d+e+f+g+h+i = 1+2+3+4+5+6+7+8+9 = 45

which is divisible by 9. You aso know that

a+b+c+d+e+f+g+h = 45 - i

is divisible by 8. i = 5 is the only digit that satisifies this condition, so i = 5. Now consider

a+b+c+d+e+f+g = 45 - i - h = 45 - 5 - h = 40 - h

This must be divisible by 7. Certainly h = 5 gives 40 - h = 35 which is divisible by 5 and no other chioce of h makes 40 - h divisible by 7. Hence h = 5.

But your requirements are that a, b, c, d, e, f, g, h and i are distinct so there is no nine digit social security number that satisfies your requirement.



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