Problem: what is the units digit of ( 63 ^ 4 ) ( 41 ^ 7 ) ( 99 ^ 9 ) ?
Rationale: "the units digit of 63 ^ 4 is the units digit of 3 ^ 4 which is 1 . the units digit of 41 ^ 7 is the units digit of 1 ^ 7 which is 1 . the units digit of 99 ^ 9 is the units digit of 9 ^ 9 which is 9 . note the pattern : 9 ^ 1 = 9 , 9 ^ 2 = 81 , 9 ^ 3 = 729 , . . . the units digit alternates between 9 and 1 . the units digit of 1 * 1 * 9 is 9 . the answer is e ."
annotated_formula: divide(add(multiply(factorial(63), factorial(4)), multiply(factorial(63), factorial(7))), 63)