More fun in 2015

Number Theory Level 3

Find the smallest positive integer $n$ such that $2015n$ has the same last three digits as $n$ .

The answer is 500.

1 solution

Jake Lai
May 21, 2015

We have $2015n \equiv 15n \equiv n \pmod{1000}$ and thus $15n-n = 14n = 1000k$ for some positive integral $k$ .

The smallest solution is then $14n = 1000k = \text{lcm}(14,1000)$ or $n = 500$ .

You can also do the following:

$\begin{aligned}2015n\equiv n\pmod{1000}&\iff 15n\equiv n\pmod{1000}\\&\iff 14n\equiv 0\pmod{1000}\\&\overset{:7}{\iff} 2n\equiv 0\pmod{5^3\times 2^3}\\&\overset{:2}{\iff} n\equiv 0\pmod{5^3\times 2^2}\\&\iff n\equiv 0\pmod{500}\end{aligned}$

The division by $7$ is valid since $\gcd(7,1000)=1$ and the division by $2$ "reduces" the modulus.

Prasun Biswas - 6 years ago

Exactly! (+1)

Otto Bretscher - 6 years ago

More fun in 2015

The answer is 500.

1 solution

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