<s>[INST] Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$ 
Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$.

I tried
$$x\equiv5\pmod6\\x\equiv4\pmod5\\x\equiv3\pmod4\\x\equiv2\pmod3$$What $x$ value?
 [/INST] The number which leaves (5, 4, 3, 2) mod (6, 5, 4, 3) is one less than the one that leaves a residue of (0, 0, 0, 0), mod (6, 5, 4, 3).  So one finds $n=\operatorname{lcm}(6,5,4,3)-1$ to get 59, which is the desired answer.
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