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A 4 digit number is randomly picked out of all the 4 digit number. Find the probability that the product of its digits is divisible by 3. If the answer can be expressed as for coprime positive integers , what is the value of ? This problem is part of the set Advanced is basic
Note: 012 is a two digit number, not a three digit number.
The answer is 232.
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1 solution
Moderator note:
Yes, probability by complement is the easiest way for this question.
It is easier to calculate the complement first. For a 4 -digit number to not have the product of its digits be divisible by 3 , none of the digits can be divisible by 3 . Now there are 6 digits not divisible by 3 , (namely 1 , 2 , 4 , 5 , 7 , 8 ), and the number of 4 -digit numbers that can be formed using just these digits is 6 4 = 1 2 9 6 .
Since there are 9 0 0 0 4 -digit numbers in all, the probability the the digits of a randomly chosen 4 -digit numbers has a "digit product" divisible by 3 is
1 − 9 0 0 0 1 2 9 6 = 1 2 5 1 0 7 , and thus a + b = 1 0 7 + 1 2 5 = 2 3 2 .