10. $a, b, c$ are all prime numbers. If $a \times b+b \times c=119$, then $a+b+c=$
【Analysis】Extract the common factor $b(a+c)=119$, find the factors of 119.
【Solution】Solution: Extract the common factor $b(a+c)=119$,
$$
119=17 \times 7=1 \times 119,
$$

Since $a, b, c$ are all prime numbers,
When $b=7$, $a+c=17$ does not satisfy the condition,
When $b=17$, $a$ and $c$ can be 2 and 5 respectively to satisfy the condition,
$$
a+b+c=17+7=24 .
$$

Therefore, the answer is: 24 .