To find the number of monomials in the polynomial resulting from the product of the two given polynomials, we need to multiply the number of monomials in each polynomial and then add them, but since the resulting polynomial will have many terms with similar exponents, we will also need to simplify it by adding the coefficients of these similar terms. 

We can start by factoring each polynomial to see its pattern:

1+t^3+t^6+...+t^30 = 1+t^3(1+t^3+...+t^27) = 1+t^3(1+t^3(1+t^3+...+t^24))

1+t^5+t^10+...+t^30 = 1+t^5(1+t^5+...+t^25) = 1+t^5(1+t^5(1+t^5+...+t^20))

Now, we can see that each polynomial has a geometric series inside it, so we can use the formula for the sum of a geometric series to simplify them:

1+t^3(1+t^3(1+t^3+...+t^24)) = 1+t^3(1-t^27)/(1-t^3) = (1-t^30)/(1-t^3)

1+t^5(1+t^5(1+t^5+...+t^20)) = 1+t^5(1-t^21)/(1-t^5) = (1-t^25)/(1-t^5)

Now, we can multiply these two polynomials:

(1-t^30)/(1-t^3) * (1-t^25)/(1-t^5) 

= (1-t^30-t^25+t^55)/(1-t^8-t^20+t^28)

Now, we need to expand the numerator and simplify by adding the coefficients of the similar terms. Since the exponents in the numerator are at most 55, we can write the polynomial as a sum of monomials with exponents from 0 to 55:

= 1 - t^8 - t^20 + t^28 - t^25 + t^3 + t^33 - t^30 - t^13 + t^11 - t^43 - t^41 + t^39 + t^37 + t^35 + t^53 + t^51 + t^49 + t^47 + t^45 + t^55

So, the number of monomials in the resulting polynomial is 21.
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