Number Theory
# Common Misconceptions (Number Theory)

Can an even number, divided by another even number, times another even number ever equal an odd number?

If "yes," then find three numbers that work.

If "no," then why not?

Note that the three even numbers can be different numbers.

For integral choices of \(x\) and \(y\), \[\text{LCM}(x, y) \leq xy.\]

Is the above statement true or false?

**Clarification:** The \(\text{LCM}\) is the Lowest Common Multiple of two numbers.

True or False: Every odd number greater than 1 is the smallest member of a primitive Pythagorean triple.

**Note:** A primitive Pythagorean triple is a triple of positive integers \((a,b,c)\) such that \(a^2+b^2 = c^2\) and \(\gcd(a,b,c) = 1.\)

\[\large\text{Irrational}^{\text{Irrational}}= \text{Irrational}\]

Is the above equation always true for irrational numbers? Each irrational number can be different.

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