A seven-digit positive integer is chosen at random. What is the probability that the number is divisible by , given that the sum of its digits is
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Note that the maximum digit sum of a -digit number is . To compute how many -digit numbers have a digit sum of , we may think about it as distributing two 's to the digits of . If they are distributed to the same digit, there are ways to do so and if they are distributed to different digits there are ways for a total of numbers with digit sum .
A multiple of must have an alternating digit sum of as well, and note that has an alternating digit sum of . Therefore, both 's must be distributed to digit places which are negative - the nd, th, and th slots. Similarly to above there are ways to do this for a final answer of
Suppose the -digit number was . Then, let and . Note that , and . Thus, , and . Thus, .
Note that if is divisible by , must also be divisible by , so , implying and . Note that , so we can choose them in
ways for . As , , so thus there are a total of such -digit numbers divisible by and sum .
To count the total number of -digit numbers with digit sum , we note that
so we can distribute them in
ways, for a probability of
The problems on this page are the property of the MAA's American Mathematics Competitions