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The Pythagorean Theorem is one of the most famous quadratic diophantine equations, which are equations with squared variables solved over the integers. How many solutions can you find?

# Level 2

Magician Mike Michael claims to know two positive whole numbers that multiply to 1000, neither of which contain the digit 0.

What is the sum of these 2 numbers?

$\large a!b! = a! + b!$

If $$a$$ and $$b$$ are positive integers that satisfy the equation above, find $$a + b$$.

If we mistakenly add numerator and denominators, we would think:

$\Large \dfrac{1}{a} + \dfrac{1}{b}=\dfrac{2}{a+b}.$

How many ordered pairs $$(a.b)$$ in the interval $$-10\leq a,b \leq 10$$ such that they satisfy the equation above.

What is the smallest positive multiple of hundred which can be expressed as the product of two consecutive integers?

How many primes are 4 less than a perfect square?

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