# The Source Code

**Number Theory**Level 5

Colter is a designated spy on a mission. His mission is to dismantle a suitcase bomb on a train to Chicago. In order to do that, he has to break *The Source Code*. The Source Code is a \(2015\times 2015\) table with buttons labelled from 1 to \( 2015^2 \) in a snake-like fashion (see below for details). The code can be broken if and only if Colter has pushed all the button whose number \(x\) satisfies the following conditions:

- \(a\) is adjacent to \(x\) from the top, and \( a \equiv 1 \pmod{2} \)
- \(b\) is adjacent to \(x\) from the right, and \( b \equiv 2 \pmod {3} \)
- \(c\) is adjacent to \(x\) from the bottom, and \( c \equiv 3 \pmod{4} \)
- \(d\) is adjacent to \(x\) from the left, and \( d \equiv 4 \pmod{5} \).

How many button numbers \(x\) does Colter have to push in order to break The Source Code?

**Clarifications**:

- Snake like fasion: For a \(n\times n\) table \((n>2)\) containing \(n^2\) buttons, each marked by a natural number from \(1\) to \(n^2\), the order the numbers are written is as follows: On the first line, the numbers \(1,2,\ldots,n\) are in ascending order, whereas on the next line \(n+1,\ldots,2n\) are in descending order. Here is an example of a \(3\times3\) table:

**Hint**: Chinese remainder theorem is going to be useful in this.