A number magician tells you: : Pick any positive integer and multiply it by 18. Delete one of the non-zero digits of the product. Tell me the sum of the remaining digits of the product and I will tell you what digit you deleted." Explain this trick.
Based on the property that for any number multiplied by 9 the sum of the digits in the answer must be a multiple of 9
since 18 = 2x9, this still holds true.
e.g. 23x18 = 414 and 414 is divisible by 9
suppose we had deleted a 4, the sum would now be 5. How much to the next multiple of 9 ?? clearly 4, which was the number deleted.
e.g. #2 4963x18 = 89334
suppose we delete the 8, the sum of the remaining digits is 19.
What is the next multiple of 9 after 19 ?
it is 27
how far from 19 to 27 ?
8 -- which is the number we deleted.