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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