FILE No.168Unsolved
In the year 21XX, phone numbers are 10 digits. Her number is a 10-digit number that uses each of 0 through 9 exactly once, and its first n digits form a multiple of n. (The first 2 digits are a multiple of 2, the first 3 digits are a multiple of 3, and so on, up to the whole 10 digits being a multiple of 10.)
Which is her phone number?
If you get stuck, open the hints one at a time, in order.
What are the last digit and the 5th digit?
What digits go in the even positions?
The first 7 digits form a multiple of 7.
Which is her phone number?
Choose the answer closest to the truth.
3816547290
The answer is "3816547290."
The last digit is 0 (multiple of 10), and the 5th digit is 5 (multiple of 5). Even digits go in the even positions (2nd, 4th, 6th, and 8th). Narrowing with this and the conditions for multiples of 3, 4, 6, 8, and 9, then checking that the first 7 digits form a multiple of 7, only 3816547290 remains.
For 1472589630, 1836547290, and 7896543210, the first 7 digits are not divisible by 7 in any case, so they fail the condition.
Because there are 10! candidates and it becomes unmanageable.
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