The polynomial (1+t^3+t^6+...+t^30)(1+t^5+t^10+...+t^30) is the product of two geometric series. The first series has a common ratio of t^3 and the last term t^30, and the second series has a common ratio of t^5 and the last term t^30. To find the number of monomials in the polynomial after opening parentheses and adding similar terms, we need to consider the powers of t that will result from multiplying these two series.

The powers of t that will result from multiplying the first and second series are the sums of the powers of t from the two series. Let's list out the powers of t from the first series:

1 + t^3 + t^6 + ... + t^30

The exponents in this series are multiples of 3, ranging from 0 to 30. Similarly, the powers of t from the second series are multiples of 5, ranging from 0 to 30:

1 + t^5 + t^10 + ... + t^30

When we multiply these two series, the exponents of t in the resulting polynomial will be the sums of the exponents from the first and second series. We need to consider all possible combinations of exponents that can be obtained by adding a multiple of 3 from the first series to a multiple of 5 from the second series, and keep track of the resulting monomials.

Let's create a table to help us visualize this:

Exponent from first series	Exponent from second series	Sum of exponents
0	0	0
0	5	5
0	10	10
...	...	...
0	30	30
3	0	3
3	5	8
3	10	13
...	...	...
3	30	33
6	0	6
6	5	11
...	...	...
30	0	30
30	5	35
...	...	...
30	30	60
Now, we can see that the sum of exponents in the resulting polynomial will range from 0 to 60, depending on the combination of exponents from the first and second series. Therefore, there will be a total of 61 different monomials in the polynomial after opening parentheses and adding similar terms.
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