Number Theory
# Linear Diophantine Equations

\[\frac { \begin{matrix} & A & B & C & D & E \\ \times & & & & 1 & 2 \end{matrix} }{ C\quad D\quad E\quad 0\quad A\quad B } \]

Find \(\overline { ABCDE } \).

How many possible 6 digit numbers are there of the form \( N=\overline{abcabd} \) where \( a \neq 0, d \neq 0\), \(d = c + 1 \) and \( N \) is a perfect square?

**Details and assumptions**

The condition of \(a \neq 0 \) follows because we have a 6 digit number.

The condition of \( d \neq 0 \) follows because \( 0 \neq 9 + 1 \).

At my butcher shop, you can buy 2 pounds of chicken thigh and 1 pound of chicken feet for 738 cents, and 3 pounds of chicken thigh and 1 pound of chicken heart for 852 cents.

For how many ordered sets of integers \( (t, f, h ) \), with each of the element between 0 and 100 inclusive, can you determine the exact total cost of \(t\) pounds of chicken thigh, \(f\) pounds of chicken feet, and \( h \) pounds of chicken heart?

Assume that the cost per pound of any given meat is constant.

Consider the system of equations

\[ \begin{split} y & = 2{x}_{1}+{x}_{2} \\ y & = 3{x}_{2}+{x}_{3} \\ y & = 4{x}_{3}+{x}_{4} \\ y & = 5{x}_{4}+{x}_{5} \\ y & = 6{x}_{5}+{x}_{1}. \end{split} \]

If all of the variables are integers, what is the minimum positive integer value of

\[\left(\sum_{i=1}^{5}{x}_{i}\right) - y ?\]