Happy Birthday Kostya!
Today is
my friend Kostya's
birthday, and this is my problem in his honour.
Andrei and Kostya sit in different rooms. Andrei writes a number from 1 to 8, inclusive, on a piece of paper. Kostya, with no knowledge of Andrei's number, also writes a number from 1 to 8 on his piece of paper. When both numbers have been written, the two pieces of paper are connected to form a two-digit number: Andrei's number goes first, Kostya's second. What is the probability that is divisible by 9?
Details and Assumptions:
-
Both numbers can be the same.
-
Express your answer as a decimal.
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There are a total of 8 numbers to choose from.
The answer is 0.125.
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A number is divisible by 9 if the digits' sum is divisible by 9 . The largest two-digit multiple of nine is 9 9 , and this is the only one to have 1 8 as the digit sum. All the other two-digit multiples of nine have a digit sum of 9 .
Since the numbers that can be written only go up to 8 , the required probability becomes that of choosing two numbers from { 1 , 2 , . . . , 7 , 8 } that add up to 9 .
Note that every single number here has a partner to which it needs to be added to make 9 . 8 + 1 = 9 , 7 + 2 = 9 , 6 + 3 = 9 , 5 + 4 = 9 . Therefore, there is no difference which number is chosen first, since there is always a 8 1 probability that its partner will be chosen!
@Kostya Popov