Fun with Simplifying Fractions

Number Theory Level 4

The number of integers $x$ with $-2014 \le x \le 2014$ for which the fraction $\dfrac{x^2+2014}{x^2+2017}$ is already in simplest form can be expressed as $1000a+100b+10c+d$ , where $a$ , $b$ , $c$ , and $d$ are (not necessarily) distinct integers from $0$ to $9$ , inclusive. Find the value of $a+b+cd$ .

The answer is 22.

1 solution

Akshaj Kadaveru
Mar 22, 2014

The condition isn't satisfied when $\gcd(x^2 + 2014, x^2 + 2017) = \gcd(3, x^2 + 2014) > 1$ , or $\gcd(3, x^2 + 2014) = 3$ . This means $x^2 + 2014 \equiv 0 \pmod3$ or $x^2 \equiv 2 \pmod{3}$ , impossible. Therefore the condition is always satisfied for an answer of $4029 \implies \boxed{22}$ .

Yup! Hey, I commented to you a little while ago on some random post asking for advice. I guess you didn't get that. Anyways, I saw you at the Countdown Round at States, and you and Franklyn were incredible. I also saw you at Nationals last year. It's kind of cool to see somebody on Brilliant who I've actually seen in real life. Regardless, I really want to know how you became so good at math. I mean, sure you can practice for hours and hours, and learn a thousand theorems and formulas, but how do you really get to the point were you can just solve any problem you want? Unfortunately, despite being the 18th most followed person on Brilliant (which is pretty awesome), I didn't do so well at States. Can you recommend any good resources or methods for learning? Also, what's your "story", like, did you start math at a really young age and what math class are you in and stuff like that. Thanks! :D

Finn Hulse - 7 years, 2 months ago

Fun with Simplifying Fractions

The answer is 22.

1 solution

0 pending reports