NMTC Practice Part 3

Number Theory Level 3

Find the positive integer $\color{#20A900}{n}$ such that $\color{#69047E}{\frac{n}{810}=0.d25d25d25d25d25......}$

[This problem is not mine. $d$ is a digit.]

1 solution

Mathh Mathh
Aug 20, 2014

$0.\overline{\text{d}25}=\frac{\overline{\text{d}25}}{999}=\frac{n}{810}$

$\stackrel{\times 29970}\implies 30(100d+25)=3000d+750=37n$

$\implies 3d+10\equiv 0\pmod{37}, 0\le d\le 9$

$\implies d=9\implies \boxed{n=750}$

It can be done without modular arithmetic

Chirayu Bhardwaj - 5 years, 3 months ago

Modular arithmetic is nothing difficult, anyways. It is just an intuition put rigorously.

Shourya Pandey - 5 years, 1 month ago