# Understand the problem

**How many four-digit numbers have this property?**

##### Source of the problem

Israel 2014, Problem 4

##### Topic

Combinatorics, Number Theory

##### Difficulty Level

6/10

##### Suggested Book

Excursion in Mathematics by Bhaskarcharya Prathistan

# Start with hints

Do you really need a hint? Try it first!

Let’s write the problem mathematically, i.e. in terms of the equations. Let’s write the condition mathematically.
Let be a four-digit number, with and , and positive integers. Then we need to have and , where are positive integers.
Now, we have to count the number of such solutions.
Observe that the sum of the digits can be at most 36. So, m = 1. Hence, a+b+c+d = 27.
This leads to .
This implies being a multiple of . Now, this implies we have quite a number of cases to investigate. Let’s do them one by one patiently.

b | c | a+d |

0 | 3,6,9 | 27,24,21,18 |

b | c | a+d |

1 | 1,4,7 | 25,22,19 |

b | c | a+d |

2 | 2,5,8 | 23,20,17 |

b | c | a+d |

3 | 0,3,6,9 | 24,21,18,15 |

b | c | a+d |

4 | 1,4,7 | 22,19,16 |

b | c | a+d |

5 | 2,5,8 | 20,17,14 |

b | c | a+d |

6 | 0,3,6,9 | 21,18,15,12 |

b | c | a+d |

7 | 1,4,7 | 19,16,13 |

b | c | a+d |

8 | 2,5,8 | 17,14,11 |

b | c | a+d |

9 | 0,3,6,9 | 18,15,15,9 |

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