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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