# What fraction of the products in the multiplication table from 1 to 10 are even numbers?

Is your gut reaction to say one half, which holds for the sums in the additon table from 1 to 10? If you think for a moment about what the rule is for determining when the product of two numbers is even, the correct answer should be apparent. The only way to get an odd product is to multiply two odd numbers, which happens a half of a half, or a quarter, of the time. Therefore ¾ of the products in a multiplication table are even.

I have found that many young students do not know the odd/even rule for multiplication, which among other things, makes it harder for them to memorize the multiplication table. I can’t think of a gut level intuitive explanation for why the rule holds. For example, you can give the following argument, whose assumptions are not that intuitively obvious: All numbers have a unique prime factorization, and a number is even when 2 is one of its prime factors. Therefore the prime factorization of the axb is odd only if 2 is not one of the prime factors of either a or b.

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## 4 Answers

I think you expressed the reason. To me, that’s easy to understand and makes intuitive sense, so once that’s understood, for me it is a “gut level” thing.

I also get at a “gut level” that the results of an integer multiplication table are *not* something I would expect to be like the whole set of integers, because it’s the results of multiplying two integers, half of which are even, and multiplication by an even integer always results in an even product.

If that’s well understood, then it seems pretty obvious that ¾ of two-integer products will be even, while half of the non-zero number line are even. (And 7/8ths of three-integer products will be even, etc…)

25%.

All of the products of an even number multiplier (50%) and all the products of an odd multiplier and an even multiplicand (25%) make up 75% of the integers ,which leaves 25% as odd integer products.

All of them.

(My answer makes sense if you read the question slightly differently without reading the context)

Here’s a quick cut at it.

Numbers 1 through10 have an equal number of even and odd numbers. They can be matched in only 4 ways

even x even = even

odd x even = even

even x odd = even

odd x odd = odd Therefore: 25%

Off the top of my head i can’t think of any exceptions.

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