If the number so formed is divisible by 7, so is the original. Hence the result. A better way of showing this is to use modular arithmetic. This easy test gives a good reason we should work in base 12 rather than base 10, since we then have an easy test for 11 and 13, rather than 9 and 11 and we lose those nasty recurring decimals when dividing by 3.
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LOOK :. Take 34 for example. Take a peek at So the digit sum of 98 is 8. Take 3, So the digit sum of 3, is 5. Sometime during the development of this condensation trick, someone, bless their soul, found out that adding 9, or any combination of digits which add to nine, does not influence a digit sum. Now let's add the digit 9 to make Adding that 9 did not change the digit sum. Quite frankly, they found that you can add as many nines or, combinations of numbers that add to nine, as you want, and it wont change the digit sum.
Take a look at , Again we arrive at a digit sum of 7. HERE you try a few. Remember, any nine or combination of numbers which add to nine, can be cast out or thought of as ZERO. Click the?
Whenever you add up a column of figures to get a correct sum, you will find that the digit sums will form a correct addition too! This allows you to condense even the most horrible numbers into little, baby, single digits and then work with the little, baby, digits to see if your calculations are correct. Just look how easy the check to this horrid problem becomes. The answer , also has a digit sum of ZERO, so your problem is correct! Quoting the author of this magical book of math shortcuts,.
The longer and more complicated the addition, the more time is saved. Furthermore, many people find it entertaining to hunt down the 9's and watch the digit sums add up correctly and, after learning the method, check their addition for the FUN OF IT. Without a doubt, the most important thing to know about casting out nines is how to use it.
So if any of the things I mentioned earlier—stuff like check digits and using casting out nines to check addition, subtraction, multiplication, and division—are unfamiliar to you, I encourage you to go back and take a look at the two earlier articles in this series. We just add up all the digits in numbers and compare them and somehow that tells us if the answer can be correct…kind of crazy!
Which inevitably leads us wonder: Why does casting out nines work? If we add up those two remainders—does anything seem familiar yet? Again, does anything seem familiar about this?
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