Taxi!

Number Theory Level 3

$\begin{cases}{a^3=b^3+c^3-d^3} \\ {a \ \ =b+c-7d}\end{cases}$

Positive integers $a,b,c,d$ satisfy the above system of equations. Find the value of $a+b+c+d$ .

-32

-23

32

23

2 solutions

Shivamani Patil
Oct 27, 2014

Logical answer :

as $a=b+c-7d$

we get $a+b+c+d=b+c-7d+b+c+d=2b+2c-6d=2(b+c-3c)$

As all are positive integers and notice $a+b+c+d$ has $2$ as factor.

So we have answer as $32$

Manuel Kahayon
Jan 23, 2016

The title of the problem is actually referring to taxicab numbers which can be expressed as the sum of two distinct pairs of cubes.

The most famous of these numbers is $1729$ , which can be expressed as either $10^3 +9^3$ or $12^3+1^3$ .

So, we get $10+12+9+1 = 32$ .