Deducing the marbles in each bag!

Let's examine the first statement:

The total marbles in 3 bags $(1), (2), (3)$ are three times as many as the marbles in bag $(4)$ .

Therefore, the total marbles in 4 bags $(1), (2), (3), (4)$ are four times as many as the marbles in bag $(4)$ .

The total marbles in 5 bags are $23+25+36+37+50=171$ , which has a remainder of 3 when divided by 4.

Examining the marbles in each bag, the only number of marbles which has a remainder of 3 when divided by 4 is $23$ .

Hence bag $(5)$ has 23 marbles, and bag $(4)$ has $\frac{171-23}{4} = 37$ marbles.

Therefore, the remaining 3 bags $(1),(2),(3)$ have either 25, 36 or 50 marbles.

The number of marbles in bag $(2)$ is an odd number.

Using the second statement, it's obvious that bag $(2)$ has $25$ marbles.

Bag $(1)$ has more marbles than bag $(3)$ .

Using the third statement, we can deduce that bag $(1)$ has 50 marbles and bag $(3)$ has $\boxed{36}$ marbles.

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Bonus: $\text{Conclusion: }$ Bag $(1)$ has 50 marbles, bag $(2)$ has 25 marbles, bag $(3)$ has 36 marbles, bag $(4)$ has 37 marbles and bag $(5)$ has 23 marbles.