Medium #3
Number Theory
Level
4
An oddie number is a -digit number with all three digits odd. The number of oddie numbers divisible by is
The answer is 41.
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1 solution
Take a three-digit number ABC. To become divisible by 3, A + B + C must be a multiple of 3. And by the condition, all digits must be odd so (1, 3, 5, 7, 9) are only numbers for A,B, and C.
Since A + B + C is congruent to 0 (mod 3) The digits to chosen are (1,0,2,1,0) mod 3. There are four cases to exhibit:
Case 1: When (0,0,0) mod 3, we can choose 8 ways. (e.g. 333, 339, 393, 933, 399, 939, 993, 999)
Case 2: When (1,1,1) mod 3, there are also 2 digits to choose in between so there are also 8 ways.
Case 3: When (2,2,2) mod 3, there is only one number that satisfies this and this is 555.
Case 4: When (1, 0, 2) or any permutation, we have 4 ways to choose the distinct digits but there are 6 possible ways to arrange the digits, hence there are 24 ways to do so.
Hence, summing, we have 41 ways.