66 = 2 x 3 x 11
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409,457 = 71 x 73 x 79
2 x 3 x 3 x 7 = 126 =2*7*3*3
Yes it is called the fundamental theorem of arithmeticand it says that every whole number greater than one, the natural numbers, can be written as a unique product of primes. Dr. Chuck Mathdoc
sqrt 66 = sqrt (2 x 3 x 11) = sqrt 2 x sqrt 3 x sqrt 11. All of these are irrational so their product is also.
Factors are whole numbers that divide exactly into a whole number. These numbers have no remainder. Primes, on the other hand, have exactly two factors which are 1 and itself.