They are each missing something

Algebra Level 1

{a×b×c=15b×c×d=30c×d×a=10d×a×b=6 \large \begin{cases}{ \color{#3D99F6}a \times \color{#624F41}b \times \color{#20A900}c = \color{#BA33D6}{15}} \\ {\color{#624F41}b \times \color{#20A900}c\times \color{#D61F06}d = \color{#BA33D6}{30}} \\ {\color{#20A900}c\times \color{#D61F06}d \times \color{#3D99F6}a = \color{#BA33D6}{10}} \\ {\color{#D61F06}d \times \color{#3D99F6}a \times \color{#624F41}b = \color{#BA33D6}{6}}\end{cases}

Given a,b,c,a, b, c, and dd are four distinct natural numbers that satisfy the system of equations above.

Determine the value of a+b+c+da+b+c+d.

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