# Of 2005 integers whose product is even, at most how many can be odd?

#### Frustration

How would you solve this? The question doesn't specify which integers, but my mentor tells me there is a solution.
Could someone please reasonably explain this to me?

Thanks!

#### Shakarri

Look at the problem the other way round. Of 2005 integers whose product is even, at least how many must be even?

• 1 person

#### JeffM

Shakarri gave you a neat way to think about this problem. Here is another way. Whenever you are stuck on a problem, it frequently helps to think about simpler problems of the same type.

If the product of two integers is even, what is the maximum number that can be odd.

If the product of three integers is even, what is the maximum number that can be odd.

Any thoughts now?

• 2 people

#### Frustration

when you put it that way, shak, it is still difficult for me to understand

I still don't understand the answer to the original question though, jeff.

#### JeffM

when you put it that way, shak, it is still difficult for me to understand

I still don't understand the answer to the original question though, jeff.

You got the correct answers to the simpler problems.

So if the product of TWO integers is even, at most ONE integer can be odd, which means least ONE integer must be even.

So if the product of THREE integers is even, at most TWO integer can be odd, which means at least ONE integer must be even.

Why must at least one integer be even in those two simple cases? Does that reason generalize?

• 1 person

#### Frustration

Yay, I got them correct! ------
Hmm, one must be even because it needs to balance out?
The pattern that I'm getting is that you basically subtract one from original number?
2004?

#### JeffM

Yes. The product of n odd numbers is an odd number. The product of an odd number and an even number is even. So you only need 1 even number as a factor of a product of integers for the product to be even. That was the clue that shakarri gave you. So you can multiply 2004 odd integers together, getting an odd number. When you multiply that odd number by an even number, you get an even number. For a product of n integers to be even, you can have at most n - 1 odd factors because at least one factor must be even.

Remember this trick of trying simpler problems of the same type when you are stuck. It doesn't always work, but if you are stuck, you need things to try.

• 1 person