# . If the product of five integers is negative, then, at most, how many of the five integers would be negative?

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1
Feb 8, 2018

5

#### Explanation:

When we have a positive number multiplied by a negative number, we get a negative number. For instance:

$- 3 \times 2 = - 6$

And when we have two negatives multiplying each other, we get a positive number:

$- 3 \times - 2 = 6$

And so with five integers multiplying each other, we'll get a product that is negative if we have an odd number of negative numbers:

$- 1 \times 1 \times 1 \times 1 \times 1 = - 1$

$- 1 \times - 1 \times - 1 \times 1 \times 1 = - 1$

$- 1 \times - 1 \times - 1 \times - 1 \times - 1 = - 1$

And we'll get an even number if we have an even number of negative numbers:

$- 1 \times - 1 \times 1 \times 1 \times 1 = 1$

$- 1 \times - 1 \times - 1 \times - 1 \times 1 = 1$

And so the maximum number of negative numbers we can have multiplying each other is 5.

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#### Explanation

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1
Feb 8, 2018

If all $5$ were negative the answer would be negative,

#### Explanation:

A negative number multiplied by a negative number gives a positive answer.

As long as there is an ODD number of negatives being multiplied, the answer will be negative.

So, in this case, if $1$ or $3$ or $5$ of the numbers were negative, then the answer would be negative.

If $2$ or $4$ of them are negative, the answer will be positive,

Consider the examples below using just ones.

$- 1 \times 1 \times 1 \times 1 \times 1 = - 1$

$- 1 \times - 1 \times 1 \times 1 \times 1 = + 1$

$- 1 \times - 1 \times - 1 \times 1 \times 1 = - 1$

$- 1 \times - 1 \times - 1 \times - 1 \times 1 = + 1$

$- 1 \times - 1 \times - 1 \times - 1 \times - 1 = - 1$

The order of the negative numbers makes no difference.

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