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"your finger can provide an (n-1) multiplication table in base n, 2<=n<=10 as follow:

hold n fingers in front of you. to multiply (n-1) by k in base n, lower the kth finger from the left. your answer is (ab)in base n or (b) in base n if a=0, where ab is a string, a is the number of fingers to the left of the finger you lowered, and b is the number of fingers to the right."

explain why this works without listing all possible cases.