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?

Given
$x=6a+5=6(a+1)-1$
$x=5b+4=5(b+1)-1$
$x=4c+3=4(c+1)-1$
$x=3d+2=3(d+1)-1$
therefore x will be of the form $(\text{L.C.M(3,4,5,6)}k-1)$ or,
$x=60k-1$ for some $k$.
Can you guess that $k$?

ANSWER:$k=1$, or $x=59$

