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Develop the equation of the line which connects the two oblique vertices $\;\{(0, a) \text{and} (b, 0)\}.\quad$ It is assumed that these are lattice points, of course.

Then solve as a linear Diophantine equation, yielding $\;\{x = c + dt;\quad y = e + ft\}.$

Then solve the two inequations $\;\{c + dt \ge 0 \; \text{and} \; e + ft \ge 0\}.$

This should yield a range $\;[g, h]\quad$ of valid values for $\;t.\quad$ Each of these values, substituted into the two expressions for $\;x\quad$ and $\;y,\quad$ will give you a lattice point of the hypotenuse.

The number of these (the question you asked) will be
max(g, h) - min(g, h) + 1.

Develop the equation of the line which connects the two oblique vertices $\;\{(0, a) \text{and} (b, 0)\}.\qquad\qquad$ It is assumed that these two points are lattice points, of course.

Then solve as a linear Diophantine equation, yielding $\;\{x = c + dt;\quad y = e + ft\}.$

Then solve the two inequations $\;\{c + dt \ge 0 \; \text{and} \; e + ft \ge 0\}.$

This should yield a range $\;[g, h]\quad$ of valid values for $\;t.\quad$ Each of these values, substituted into the two expressions for $\;x\quad$ and $\;y,\quad$ will give you a lattice point of the hypotenuse.

The number of these (the question you asked) will be
max(g, h) - min(g, h) + 1.

Develop the equation of the line which connects the two oblique vertices {(0, a) and (b, 0)}. It is assumed that these are lattice points, of course.

Then solve as a linear Diophantine equation {x = c + dt; y = e + ft}.

Then solve the two inequations {c + dt >= 0 and e + ft >= 0}.

This should yield a range [g, h] of valid values for t. Substituting each of these values into the two expressions for x and y will give you a lattice point of the hypotenuse.

The number of these (the question you asked) will be
max(g, h) - min(g, h) + 1.

Develop the equation of the line which connects the two oblique vertices $\;\{(0, a) \text{and} (b, 0)\}.\quad$ It is assumed that these are lattice points, of course.

Then solve as a linear Diophantine equation, yielding $\;\{x = c + dt;\quad y = e + ft\}.$

Then solve the two inequations $\;\{c + dt \ge 0 \; \text{and} \; e + ft \ge 0\}.$

This should yield a range $\;[g, h]\quad$ of valid values for $\;t.\quad$ Each of these values, substituted into the two expressions for $\;x\quad$ and $\;y,\quad$ will give you a lattice point of the hypotenuse.

The number of these (the question you asked) will be
max(g, h) - min(g, h) + 1.

Develop the equation of the line which connects the two oblique vertices {(0, a) and (b, 0)}. It is assumed that these are lattice points, of course.

Then solve as a linear Diophantine equation {x = c + dt; y = e + ft}.

Then solve the two inequations {c + dt >= 0 and e + ft >= 0}.

This should yield a range [g, h] of valid values for t. Substituting each of these values into the two expressions for x and y will give you a lattice point of the hypotenuse.

Develop the equation of the line which connects the two oblique vertices {(0, a) and (b, 0)}. It is assumed that these are lattice points, of course.

Then solve as a linear Diophantine equation {x = c + dt; y = e + ft}.

Then solve the two inequations {c + dt >= 0 and e + ft >= 0}.

This should yield a range [g, h] of valid values for t. Substituting each of these values into the two expressions for x and y will give you a lattice point of the hypotenuse.

The number of these (the question you asked) will be
max(g, h) - min(g, h) + 1.