### Brain Teasers

# Multiple Divisors

Professor Abacus is looking for a number. The number must have 10 digits. It must contain all 10 numerals (0 through 9, once each). It must also be evenly divisible by all integers 2 through 18. Find one of the four possible solutions.

### Answer

First, use prime factorization to determine the lowest common multiple (LCM).2 = 2

3 = 3

4 = 2 * 2

5 = 5

6 = 2 * 3

7 = 7

8 = 2 * 2 * 2

9 = 3 * 3

10 = 2 * 5

11 = 11

12 = 2 * 2 * 3

13 = 13

14 = 2 * 7

15 = 3 * 5

16 = 2 * 2 * 2 * 2

17 = 17

18 = 2 * 3 * 3

LCM = 2 * 2 * 2 * 2 * 3 * 3 * 5 * 7 * 11 * 13 * 17 = 12252240.

The smallest 10-digit number that is a multiple of the LCM is 1004683680 (12252240 * 82). The largest 10-digit number that is a multiple of the LCM is 9997827840 (12252240 * 816). The solution must be somewhere between these two numbers.

To solve, multiply the LCM by each of the integers 82 through 816. The four solutions are:

2438195760 = 12252240 * 199

3785942160 = 12252240 * 309

4753869120 = 12252240 * 388

4876391520 = 12252240 * 398

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