How the #uck does this work ?
Discussion
Not quite. The answer is always exactly divisible by 9 (and all multiples of 9 equate to the same symbol).
It works because the digits of any multiple of 9 always sum to a multiple of 9.
The number you thought of can be represented as 9a + b with 0 < b < 10. Therefore the sum of your number's digits can be represented as 9c + b (because 9a is a multiple of 9 and thus the sum of it's digits is 9)*. So the subtraction requested by the trick is (9a + b) - (9c + b) = 9(a-c) which is a multiple of 9.
Not a rigorous proof, but what the hell.
* The not so obvious case here is where adding b to 9c increments the "tens" column. This gives us a "+1" but we got that from doing a "-10" in the units which nets out at "-9" which again is divisible by 9.
It works because the digits of any multiple of 9 always sum to a multiple of 9.
The number you thought of can be represented as 9a + b with 0 < b < 10. Therefore the sum of your number's digits can be represented as 9c + b (because 9a is a multiple of 9 and thus the sum of it's digits is 9)*. So the subtraction requested by the trick is (9a + b) - (9c + b) = 9(a-c) which is a multiple of 9.
Not a rigorous proof, but what the hell.
* The not so obvious case here is where adding b to 9c increments the "tens" column. This gives us a "+1" but we got that from doing a "-10" in the units which nets out at "-9" which again is divisible by 9.
that is very good.
pretty simple maths too
AB in decimal notation is the same as
10A + 1B, therefore
10A+B - (A+B) = 10A + B - A - B
= 9A
which is a multiple of 9
( in case anyone was nerdy like myself and wandered why it worked )
edited to say beaten to it by V8 archie
>> Edited by ledger on Thursday 30th June 16:30
pretty simple maths too
AB in decimal notation is the same as
10A + 1B, therefore
10A+B - (A+B) = 10A + B - A - B
= 9A
which is a multiple of 9
( in case anyone was nerdy like myself and wandered why it worked )
edited to say beaten to it by V8 archie
>> Edited by ledger on Thursday 30th June 16:30
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