Altitude Projection Formula — MathGoldMedalist

a=14

x=\frac{14^2+13^2-b^2}{28}

x=\frac{196+169-b^2}{28}

x=\frac{365-b^2}{28}

365-b^2\equiv0\pmod{28}

b^2\equiv365\equiv1\pmod{28}

b^2\equiv1\pmod4 \Longrightarrow b\equiv \underset{\times}{0}, \underset{\checkmark}{1}, \underset{\times}{2}, \underset{\checkmark}{3} \pmod4

b^2\equiv1\pmod7 \Longrightarrow b\equiv \underset{\times}{0}, \underset{\checkmark}{1}, \underset{\times}{2}, \underset{\times}{3}, \underset{\times}{4}, \underset{\times}{5}, \underset{\checkmark}{6} \pmod7

So,

b\equiv1,3\pmod4

b\equiv1,6\pmod7

Therefore

b\equiv1,15,13,27\pmod{28}

By Triangle Inequality

b<13+14=27

So,

\max\{b\}=\boxed{15}

Check x:

x=\frac{365-15^2}{28}=\frac{365-225}{28}=\frac{140}{28}=5