A positive integer is called sparkly if it has exactly 9 digits, and for any n between 1 and 9 (inclusive), the nth digit is a positive multiple of n. How many positive integers are sparkly?
Problem
Source: 2019 NZ Mathematical Olympiad Round 2
Tags: combinatorics
naman12
17.12.2019 07:32
The first digit has $9$ options. The second has $5$. The third has $4$. The fourth has $3$. The rest have $2$. The answer is $9\cdot 5\cdot 4\cdot 3\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=17280.$
Sudoku11
17.12.2019 08:39
naman12 wrote: The first digit has $9$ options. The second has $5$. The third has $4$. The fourth has $3$. The rest have $2$. The answer is $9\cdot 5\cdot 4\cdot 3\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=17280.$ I think, the second has 4 (it is 2, 4, 6, 8). The third has 3 (3, 6, 9). The fourth has 2 (4, 8). The rest has 1.
naman12
17.12.2019 08:47
We can include 0, no?
Sudoku11
18.12.2019 04:52
naman12 wrote: We can include 0, no? I think, the number 0 is not satisfied, because number 0 is not positive.