Abstract
Summary Notice that the square of 9376 is 87909376 which has as its rightmost four digits 9376. To generalize this remarkable fact, we show that, for each integer , there exists at least one and at most two positive integers x with exactly n-digits in base 10 (meaning the leftmost or nth digit from the right is nonzero) such that squaring the integer results in an integer whose rightmost n digits form the integer x. We then generalize the argument to prove that, in an arbitrary number base with exactly m distinct prime factors, an upper bound is and a lower bound is for the number of such n-digit positive integers. For , there are exactly solutions, including 1 and excluding 0.
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