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Q: How do “Numerology Math Tricks” work? (adding digits and tricks with nines)

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For example,An example for multiplication:So we’ve gotandBut when you do math mod nine you get one more, strange property: a number mod 9, is equal to the sum of its digits mod 9. All the fancy tricks you may have learned in grade school involving nines all boil down to this property. It doesn’t matter if you look at x or the sum of x’s digits (don’t change mod 9). : In steps 2 and 3 you’re forcing your friend to create a new number X, such that [X]= [0]. Moreover (using that same rule again) the sum of X’s digits must also be a multiple of 9.