Problem: for any positive number x , the function [ x ] denotes the greatest integer less than or equal to x . for example , [ 1 ] = 1 , [ 1.367 ] = 1 and [ 1.999 ] = 1 . if k is a positive integer such that k ^ 2 is divisible by 45 and 80 , what is the units digit of k ^ 3 / 4000 ?
Rationale: "factorization of 45 = 3 * 3 * 5 factorization of 80 = 5 * 2 ^ 4 means the smallest value of k is \ sqrt { 3 ^ 2 * 5 * 2 ^ 4 } = 3 * 5 * 2 ^ 2 k ^ 3 = 3 ^ 3 * 5 ^ 3 * 2 ^ 6 * x = 3 ^ 3 * 2 * 4000 * x where x can be any integer k ^ 3 / 4000 = 3 ^ 3 * 2 * x = 54 x which obviously has different units digit depending on x answer : d"
annotated_formula: divide(multiply(multiply(multiply(3, 2), multiply(3, 2)), multiply(3, 2)), const_4)