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The problem: Find the smallest positive integer $x$ such that

$x$ divided by $4$ has remainder $1$

$x$ divided by $5$ has reminder $2$

$x$ divided by $6$ has remainder $3$

Now, my first idea was to add to each divisor its the remainder and multiply the quantities obtained.

$x=(4+1)(5+2)(6+3)=315$

But $315$ does not satisfy all the conditions above, and I don't know how to get the smallest integer that satisfies the conditions. Any help?

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  • $\begingroup$ Are you familiar with the congruence relations on the integers? $\endgroup$ – Luca Bressan Sep 9 '16 at 17:29
  • $\begingroup$ Yes but im not fan of them $\endgroup$ – Peterix Sep 9 '16 at 17:30
  • $\begingroup$ $315$ is neither the smallest number that works, nor is it even one such number. $315$ leaves a remainder of $0$ when divided by $5$ ... $\endgroup$ – Zubin Mukerjee Sep 9 '16 at 17:33
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If $x \equiv 1\pmod{4}$, then $$x \equiv 1,5,9,13,17\pmod{20}$$ Of these, only $17$ is also $2 \pmod{5}$. Therefore $$x \equiv 17 \pmod{20}$$

Next, the LCM of $20$ and $6$ is $60$.

If $x \equiv 17 \pmod{20}$, then $$x \equiv 17, 37, 57 \pmod{60}$$

Of these, only $57$ is $3\pmod{6}$. This means

$$x \equiv 57\pmod{60}$$

This means $\boxed{57}$ is the smallest $x$ that satisfies your conditions.

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  • $\begingroup$ Note that this method also finds all $x$ that satisfy your conditions. Simply add any multiple of $60$ to $57$ to get other values of $x$ that will work. $\endgroup$ – Zubin Mukerjee Sep 9 '16 at 17:33
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You may have solved problems of the kind: Find x such that it leaves a remainder 3 when divided by 4, 5 or 6.

In the above scenario you first find out a number which divisible by each of 4, 5 and 6 (lcm or it's multiple) and then you add the remainder 3 to that number. You can find infinitely many such numbers as there are infinite number of multiples of $lcm(4,5,6)$.

Here notice that $4-1 = 3$, $5-2 = 3$ and $6-3 = 3$. So basically we are solving the exactly same problem here, we have just changed the remainder from $3$ to $-3$.

As $lcm(4,5,6) = 60$, such numbers are of form $60n - 3$, for any integer $n$.

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