The positive integers $r, s$ and $t$ have the property that $r \times s \times t=1230$. What is the smallest possible value of $r+s+t$ ?
(A) 51
(B) 52
(C) 54
(D) 58
(E) 53
First, we factor 1230 as a product of prime numbers:

$$
1230=2 \times 615=2 \times 5 \times 123=2 \times 5 \times 3 \times 41
$$

We are looking for three positive integers $r, s, t$ with $r \times s \times t=1230$ and whose sum is as small as possible. We note that all of the possibilities given for the smallest possible value of $r+s+t$ are less than 60 .

Since 41 is a prime factor of 1230 , then one of $r, s, t$ must be a multiple of 41 .

Since all of the possibilities given for the minimum sum $r+s+t$ are less than 60 and the second smallest multiple of 41 is 82 , then the multiple of 41 in the list $r, s, t$ must be 41 itself.

Thus, we can let $t=41$.

Now we are looking for positive integers $r$ and $s$ with $r \times s=2 \times 3 \times 5=30$ and whose sum $r+s$ is as small as possible. (Since $t$ is now fixed, then minimizing $r+s+t$ is equivalent to minimizing $r+s$.)

The possible pairs of values for $r$ and $s$ in some order are 1,30 and 2,15 and 3,10 and 5,6.

The pair with the smallest sum is 5,6 .

Therefore, we set $r=5$ and $s=6$. This gives $r+s+t=5+6+41=52$.

ANSWER: (B)