MHB Finding Minimum Value of $n$ for Given Sum and Product

[Albert1]

$n\in N,\,\,and \,\, a_1,a_2,a_3,-------,a_n\in Z$
$if \,\, a_+a_2+a_3+-----+a_n=a_1\times a_2\times a_3\times------\times a_n=2006$
$find \,\, min(n)$

 

[lfdahl]

My attempt:

Spoiler

The prime factorization of $2006$ is $2\cdot17\cdot59$. This leaves very few possibilities to express $2006$ as a product of integers:(i). $2\cdot17\cdot59$. Sum of factors: $78$.

(ii). $34\cdot59$. Sum of factors: $93$.

(iii). $17\cdot118$. Sum of factors: $135$.

(iv). $2\cdot1003$. Sum of factors: $1005$.

(v). $1 \cdot 2006$. Sum of factors: $2007$.

In order to obtain the expression:
$a_1\cdot a_2…\cdot a_n = a_1+a_2+…+a_n = 2006$, we only have the factors mentioned in (i)-(v) and the neutral factor $\pm 1$ to fill out with. So, the task is to use a factorization, which involves the fewest number of $1$´es.

Keeping this in mind, the lowest number of factors/terms is obviously obtained, when the sum of factors is largest (case (v).). Consequently, we are aiming for the identity:

$(-1) + 1 + (-1) + 1 + 2006 = (-1) \cdot 1 \cdot (-1) \cdot 1 \cdot 2006$

Hence, the minimum number of factors/terms is $n_{min} = 5$.

P.S.: Case (iv). would imply $n = 1003$.