hey, i was wondering where the nine test comes from?

Answer

The nine-trial is already mentioned in the 3rd century AD by the Roman bishop Hippolytos. It also appears more and more in later writings, both here in Europe and in the East, namely in the twelfth century in India and in Arabia. After all, know that the Arabs were very strong in algebra. By the way, our word “algebra” comes from Arabic, from the word “al Jabr” which means completion). Did those peoples pass it on to each other? That may be true.

Be aware that the nine test does not provide complete certainty: if it does not come out, you have definitely made a mistake, but if it does come out, your result may still be wrong.

Why does the 9 trial work?

Well, take 31 for example, the remainder after dividing by 9 is 4. We say “31 modulo 9 is 4”
The fun part is now:
If AB = C then ( A modulo 9 ) . ( B modulo 9) equal to ( C modulo 9)
! if ( A modulo 9 ) . ( B modulo 9 ) is even greater than 9, you must also apply modulo 9 to that too

Example : 39 x 58 = 2262 and this is 251 x 9 + 3. So 2262 modulo 9 = 3
but 39 x 58 = ( 4 x 9 + 3 ) x ( 6 x 9 + 4 ) Now just take 3 X 4 = 12, and taking this again modulo 9 gives 3
=> the same 3 as with 2262 modulo 9.

why is this so?

You can write any number as a 9-fold + be modulo-9 :
sdus : write : A = 9-fold + a, B = 9-fold + b
so AB =( a 9-fold + a) . ( a 9-fold + b) = all 9-folds + ab
(possibly also further simplify ab first as a 9-fold + a modulo)
So, in the end, the modulo of the product corresponds to the product of the modulos.

Use in the 9 trial :

We use those modulo-9s as a test in the 9-test because they are easily obtained small numbers that are nevertheless representative of the full number, because we calculate the modulo as the sum of all digits (see below why). An error in the number will therefore almost certainly also give an error in the sum of its digits, and therefore an error in the modulo. Therefore, the nine test, which simply works with the sum of the digits, is nevertheless a useful test for the product of the whole numbers.

One more problem : How do you easily find the modulo-9 of a number?

Well : the sum of the digits of a number sytem corresponds to the modulo 9 Why ?
This is very easy to see: suppose we subtract 9 from a number. What is the effect on the sum of the figures?
Subtracting nine is the same as subtracting ten first, then adding 1.
Take for example 247 (the sum of the digits is 13) : we first do -10, this decreases the tens digit by 1, so the 4 becomes a 3. Then we do plus 1, this increases the digit of the units by 1, the 7 now becomes an 8. The total sum of the digits therefore remains the same because with the tens you drop by 1 and with the units you increase by 1.
What exactly is a modulo-9? Just what happens if you subtract the maximum number of times 9, but we have just seen that this has no effect on the total sum of the numbers. So the modulo-9 corresponds to the total sum.

Eg : 259 = 28 * 9 + 7, so 259 modulo 9 = 7
and indeed 2 + 5 + 9 = 16 and then again 1 + 6 = 7 !