My lock's combination is the smallest positive integer such that is a multiple of 3, is a multiple of 5, is a multiple of 7, is a multiple of 9, and is a multiple of 11.
What is
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Although the problem can be solved using modular arithmetic, it is shorter to try and guess the solution. The condition ' N is a multiple of 3' is redundant, being that N +3 is a multiple of 9. So we start by finding the two smallest consecutive numbers that are multiples of 5 and 7 respectively: 20 and 21 . If we add 5·7=35 to these numbers, we obtain higher consecutive numbers that are multiples of 5 and 7 respectively. After four tries, we reach a third consecutive number that is a multiple of 9: 160, 161 and 162 . Now, by adding 5·7·9=315 a few times (5), we get a fourth consecutive number that is a multiple of 11: 1735,1736, 1737 and 1738 , which is 4 units greater than the combination we are looking for: 1734 .