Did you ever notice that there are more even numbers than odd numbers in the multiplication table?

25% of whole number pairs, multiplied will give an odd answer. An even and an even make even, an even and odd make even, only odd and odd give an odd.

We learned this in 3rd or 4th Grade.

Every odd number is half of an even number.

Some even numbers are not double an odd number.

Well, yeah. 2 is the only even prime number, so every other prime number is odd, and prime numbers will not appear in a multiplication table. As a kid, I always found odd and prime numbers more interesting than even numbers.

Yes, I did notice that, and thought about why. I was also lucky enough to have parents and teachers that encouraged thinking about why the table is the way it is, and to remember rules about patterns by understanding why those patterns are there, not to just memorize the table numbers.

How to calculate the number of even numbers? I don’t remember ever being assigned that goal, and never had a reason to, but we did learn how factors work. Every other number on a number line (or axis on the mult. table) is naturally a multiple of two, and it only takes one even number in a two-number multiplication to make the result even (since that’s what even means – having a factor of 2 in it), and odd x odd is never even, so that sounds to me like it should be 75%, IF you have an even number of rows and columns, and assuming you remember and accept the traditional notion that zero is defined an even number.

We can apply the analysis done by @Smashley and @Zaku to find the proportion of numbers in the multiplication table that are divisible by 3. We again have four cases – each of the two numbers being multiplied is divisible by 3 or not divisible by 3. Again only one case gives non-divisibility. We can only end up with a number not divisible by 3 from two numbers not divisible by 3 being multiplied. Since ⅔ of numbers are not divisible by 3, that gives ⅔×⅔ = 4/9 of the table is not divisible by 3, so 5/9 of the table is divisible by 3, a little over half. Note that this works out exactly only when the number of rows and column are divisible by 3, so it is true for a 9 by 9 table or a 12 by12 table.

Working with prime numbers like 2 and 3 is relatively easy. I tried to find a systematic way of determining how many numbers are divisible by 6 in a 6 by 6 multiplication table. By enumerating all the cases for 6, 12, 18 , 24, 30 and 36, you can find that there are15 entries divisible by 6, but I can’t think of a formula for doing this.

I came up with an equation for numbers of the form pq where p and q are two different primes. It worked in two cases. I will have to test it further, I wrote a program to crank out the value. The equation for portion of numbers of the form pq is (4pq -2p -2q+1)/(pq)**2. I may check with Math Stack Exchange to see if there is a general equation for all cases.