Bunuel wrote:

What is the least possible product of 5 different integers, each of which is between –5 and 5, inclusive?

A. –1600

B. –1450

C. –1200

D. –840

E. –600

Dokami wrote:

Honestly, I don't understand this question...

The Number which we are looking for should be devisible by 5,4,3,2,1 and their negatives.

So, 3*4*5= 60 and 60 is devisible by all of the numbers and their negatives.

Why do we have to calculate this way: (-5)*5*4*(-4)*(-3) ?

And if we do so, why only with (-3) instead of (-5)*5*4*(-4)*(-3)*(3) ?

Dokami , this is really a maximize / minimize and number properties question. Think about multiplying, the product, and the number line -- not divisibility.

You wrote, "Why do we have to calculate this way: (-5)*5*4*(-4)*(-3)?"

Because to maximize the magnitude of a number, you choose the greatest factors. I'll get to your second question.

The "least possible product" means the most negative number, a big number with a negative sign.

So we want the number farthest to the left of 0 on the number line.

To make a number with a large magnitude from of a bunch of factors multiplied, it makes sense to choose the biggest factors. Integer factors from which to choose:

\(-5\leq{x}\leq{5}\) OR

_-5__-4__-3__-2__-1__0__1__2__3__4__5

I picked +5 and -5 first. They are the greatest factors, so they will maximize the product.

Then I picked the next two factors with greatest magnitude: +4 and -4

Now number properties enter**: do you pick 3, or -3?

With these four numbers, there are two positive numbers, and two negative numbers. Product?

(+)(+)(-)(-) = (+)(+) = (+) Positive.

We need a negative number to make the product negative: an odd number of negative factors makes a product negative.

You don't need to know that number property. These numbers are manageable.

(5 * 4 * -5 * -4) = 20 * 20 = 400, because (-5 * -4) = positive 20.

If you pick 3, you get (20 * 20 * 3) = 1200

Pick -3, and you get (20 * 20 * -3) = -1200

-1200 is the least possible product because it uses the largest factors, and is negative. It's as far from zero to the left as the numbers, multiplied, will get.

**Your second question: "And if we do so, why only with (-3) instead of (-5)*5*4*(-4)*(-3)*(3)[?]"

First, the prompt says only 5 numbers.

Second, the product of your numbers is positive, certainly not the "least possible product"; not the most to the left of 0.

Hope that helps.

_________________

In the depths of winter, I finally learned

that within me there lay an invincible summer.

-- Albert Camus, "Return to Tipasa"