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These are distinct points.
Thus, all 30 factor pairs produce valid lattice points.
Hence, the total number of lattice points is $ \boxed{30} $.
Question: What is the sum of all the positive divisors of $ 1440 $ that are congruent to $ 1 \pmod{4} $?
Solution: First, factor $ 1440 $:
= 144 \cdot 10 = (12^2) \cdot 2 \cdot 5 = (2^4 \cdot 3^2) \cdot 2 \cdot 5 = 2^5 \cdot 3^2 \cdot 5
We seek the sum of all positive divisors $ d \mid 1440 $ such that $ d \equiv 1 \pmod{4} $.
A divisor $ d $ satisfies $ d \equiv 1 \pmod{4} $ if it is odd (so $ d $ not divisible by 2) and $ d \equiv 1 \pmod{4} $. So restrict to odd divisors. Since $ d $ must be odd, we ignore the power of 2. So consider only divisors of $ 3^2 \cdot 5 = 45 $.
The odd divisors of 1440 are exactly the divisors of 45. List them:
1, 3, 5, 9, 15, 45