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I want to brag with the size of my lawn gnome collection to pick up ladies.

To make it easier to count them, I've tried to put them into groups of 10, then groups of 5, then groups of 4, of 3 and even in pairs of 2. Always exactly one lone gnome remained. Now I'm sick of counting them. What is the minimum number I can be sure to have?

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    $\begingroup$ Somehow, I don't think the vast majority of women are interested in the size of your lawn gnome collection... $\endgroup$
    – dcfyj
    Commented Oct 14, 2016 at 15:04
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    $\begingroup$ You obviously don't know the right women. $\endgroup$
    – RowlandB
    Commented Oct 14, 2016 at 15:18
  • $\begingroup$ This sounds like some awful question on a school test. $\endgroup$ Commented Oct 14, 2016 at 15:50
  • $\begingroup$ Just, … way too many. $\endgroup$ Commented Oct 14, 2016 at 23:34

1 Answer 1

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The number of gnomes you have must be congruent to 1 modulo 10, 5, 4, 3, and 2. By the Chinese remainder theorem, this is exactly equivalent to being congruent to 1 modulo the lowest common multiple of 10, 5, 4, 3, and 2, i.e. modulo 60. So the smallest number of gnomes you could have is

61

(assuming we exclude the trivial case where you have exactly one gnome).

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    $\begingroup$ I would be unsurprised if the "trivial case" were the one originally intended :-). $\endgroup$
    – Gareth McCaughan
    Commented Oct 14, 2016 at 15:14
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    $\begingroup$ Nah. One wouldn't impress the ladies. Being both a gnome and a lady aficionado, I know this. $\endgroup$ Commented Oct 14, 2016 at 18:39
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    $\begingroup$ @ChrisCudmore ... a "lady aficionado"? :-/ $\endgroup$ Commented Oct 14, 2016 at 19:07

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