Find the smallest natural number that leaves residues $5,4,3,$ and $2$ when divided respectively by the numbers $6,5,4,$ and $3$

Question

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?

Answer 1

HINT: Notice that your congruences are equivalent to the following ones:

$$\left\{\begin{align*} x\equiv-1\pmod6\\ x\equiv-1\pmod5\\ x\equiv-1\pmod4\\ x\equiv-1\pmod3 \end{align*}\right.$$

In other words, $x+1$ is divisible by $6,5,4$, and $3$. What’s the smallest positive integer with that property?

Answer 2

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 $x=59$

Answer 3

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.

Answer 4

The problem can be solved by Chinese Remainder Theorem (CRT) or by taking lcm of divisors. Here divisors are 3, 4, 5 and 6; and the 1 must be subtracted? can you guess why we are subtracting 1? Or you can ask me if you need any clarifications.