6. Given that the number of divisors of the positive integer $a$ is 15, and the number of divisors of the positive integer $\mathrm{b}$ is 20, and $a+\mathrm{b}$ is a perfect square, what is the smallest value of $a+\mathrm{b}$ that satisfies the conditions?

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【Answer】576 $20=2^{2} \times 5$, numbers with 20 factors include: $A^{19}$ or $A^{3} \times B^{4}$ or $A \times B^{9}$ or $A \times B \times C^{4}$. Clearly, numbers with particularly high exponents will be very large, so we should try to choose the factorization with more factors and smaller exponents.

Numbers with 15 factors, among the smaller ones, include:
$$
2^{4} \times 3^{2}=144, \quad 2^{4} \times 5^{2}=400, \quad 2^{4} \times 7^{2}=784, \quad 3^{4} \times 4=324
$$

Numbers with 20 factors, among the smaller ones, include:
$$
\begin{array}{l}
2^{4} \times 3 \times 5=240, \quad 2^{4} \times 5 \times 7=560 \\
2^{4} \times 3 \times 7=336, \\
2^{4} \times 3 \times 11=528, \quad 2^{4} \times 3^{3}=432
\end{array}
$$
(1) $144+240=384$ (discard)
(2) $144+336=480$ (discard)
(3) $324+240=564$ (discard)
(4) $144+432=576=24^{2}$ (meets the condition)

Therefore, 576 must be the smallest number that meets the condition.