Final Digit Strings of Cubes

Abstract

Let s be the string of decimal digits 31415926535 . . . 5275045519 formed by the first billion digits of zz. You have probably wondered idly whether there is an integer n such that n3 ends in s. Or, to take another example, let s be formed by concatenating the decimal digits of 9993, 9983, . . ., 23, 13, and ask the same question. The fourth annual North Central Section/MAA Team Contest [1] of November 2000, contained the following problem: Is there an integer n such that n3, in decimal form, ends in 2000 ones? The answer to all three of the above questions is affirmative. Indeed, if s is any string of decimal digits ending in 1, 3, 7, or 9, there is an integer n such that n3 ends in s, as we'll see in the next section. What about other final digit strings? If the last digit of s is anything other than 1, 3, 7, or 9, there may or may not be a cube ending in s. The precise conditions under which there is such a cube are rather interesting, and are given in three subsequent sections. From the material there you can conclude that there are cubes ending VOL. 77, NO. 2, APRIL 2004 149