Did you know this property of multiplying two numbers?
You probably already know that the last digit in the product of two numbers depends only on the last digits of the numbers being multiplied. For example, if one number ends in 7 and another in 2, when they are multiplied the last digit will be 4.
It did not occur to me until I saw it mentioned, that the same holds true for the last 2 digits, last 3 digits and so on. For example 11×13 = 143. It therefore follows that 211×513 must end with 43, which it does. 211×513=108,243.
If you write out a multiplication, you can see how this works. If you know modular arithmetic, the proof is immediate.
I can’t think of any practical use for this, but it might be a fun thing to teach in elementary school when students learn multiple digit multiplication.
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Alert! Klingon being translated into Russian, and then into the African clicking language ahead!
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I’ve always used multiplying the last 2 digits to arrive at the number. You took it further.
Interesting.
I’ve always been interested in mathematical curios like this. 143 is a good example. You can see that 143 can be calculated from 11×13 by adding the digits of the number you’re multiplying by 11 (1 and 3) and placing the sum in between those digits. This works for multiplying any two-digit number by 11. 11×23 (2+3=5, so 253). 11×71 (7+1=8, so 781). I was never taught that, but it’s a fun trick.
Similar to what you posted, there are a lot of patterns with square numbers. Square numbers can only end in 0, 1, 4, 9, 6, and 5 and it follows a specific pattern: 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, etc. Patterns occur for the last two digits of square numbers as well:
50×50=2500, 51×51=2601, 52×52=2704, 53×53=2809, 54×54=2916. Based on this pattern, what do you think 55×55 and 56×56 are without calculating them?
When I was at college, I remember a psychology major mentioning that he was taught that last rule, for reasons that escape me.
If I were to do 11×13 in my head, I would figure 13×10 = 130 +13 = 143.
We used that trick when multiplying numbers with our slide rules. The slide rule answer would get you close but the trick would add a significant digit or 2 and that would blow peoples’ minds!
—Yes….i said it. Slide rules!—-
I used to play with my dad’s slide rule.
I love slide rules. I have a Web page telling how they can be used for teaching. You can use two ordinary rulers to make a slide rule for addition. By making the scale proportional to the square of distance, you can make a slide rule for finding the third side of a right triangle. The Web page also has a slide rule for relativistic addition of velocities. I don’t know anything about relativity, but I saw an equation that could be used in a slide rule.
I don’t know how successful I was, but the slide rule is a good tool for demonstrating the important mathematical concept of isomorphism, which I attempted to do.
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