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The only cases are ones containing a product that ends in 5?

Example

193 has no digits with 2,4,6,8,0
715 also does
193 x 715 = 137,995 which also has no digits 2,4,6,8,0

Try ANY pair of 3 digits with all 1,3,5,7,9 numbers and you’ll see there’s at least 1 even-numbered digit in the multiplied product and the rare time there isn’t, 100% of those rare times it’s with a value of 5 in the 1’s digit

Which means the highest pair that doesn’t have this property seems to occur when one of the 2 numbers multiplied together is less than 10.

Example: 7 X 53 = 371

9 X 13 = 117

Once both numbers have at least 2 digits I don’t think you can preserve excluding 2,4,6,8,0 without using a multiple of 5

I don't know yet what prevents it but I tried 713 x 193 and got 137,609. Neither of the numbers that I multiplied has an even digit or a 5 and the answer has one even number (the 6). I've known about another mathematical phenomenon for years that I have never really spoken about. It is when you take any number and multiply it by itself. Then take the two numbers that are one more and less than the original number and multiply those. The end result is that the number multiplied by itself is one more than the other total. Examples: 2 x 2 = 4 which is one more than 3. 1 x 3 = 3. 5 x 5 = 25 which is one more than 24. 4 x 6 = 24.

Last edited by Pineapple674; 08-27-2022 at 03:10 PM..
Reason: Spelling error

If you look at your products of the three-digit numbers with only 1, 3, 7, 9, you will notice that the tens place (second-to-last) is even.

Why is that? In (abc) * (def), the tens place is (the ones digit of ce) + (the ones digit of bf) + (the tens digit, if any, of cf). Let's break that down.

(the ones digit of ce) and (the ones digit of bf) are clearly both odd, so their sum is even. But what about (the tens digit, if any, of cf)?

Well, the only choices are 7*3 = 21, 9*3 = 27, 7*7 = 49, 9*7 = 63, and 9*9 = 81, all of which have an even tens digit. Therefore, the overall tens digit will be even!

BUT, if either c or f is 5, (the tens digit, if any, of cf) could be odd.

It is when you take any number and multiply it by itself. Then take the two numbers that are one more and less than the original number and multiply those. The end result is that the number multiplied by itself is one more than the other total. Examples: 2 x 2 = 4 which is one more than 3. 1 x 3 = 3. 5 x 5 = 25 which is one more than 24. 4 x 6 = 24.

I like this one, too.

With algebra, . . .

Let x = that "middle number."
Then one less than x times one more than x is
(x - 1)(x + 1)
If you multiply this out, you get

x² + x - x - 1

= x² - 1

So if x = 5, you have
(5-1)(5+1) = 25-1

With geometry, . . .

Take a bunch of tiles and make a square. I'll use a five-by-five square, using 25 tiles.
Now take off a column and add a row. If you pull off a column, you are removing five tiles. But to add a row, you'll only be putting on four tiles. Now you have six rows of four, but you have an extra tile. So a 6X4 rectangle takes one fewer tiles than a 5x5 rectangle.

I realize this is the education forum, but I can't tell if this post is homework help or just something that mathy people think is interesting. I do not think this is the place for it.

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