In this exercise, we generalize the rule given in Exercise to find the squares of integers with final base 2B digit B, where B is a positive integer. Show that the base 2B expansion of the integer (anan−1 … a1a0)2Bstarts with the digits of the base 2B expansion of the integer (anan−1 … a1) 2B [(anan−1 … a1) 2B+1] and ends with the digits B/2 and 0 when B is even, and the digits (B − 1)/2 and B when B is odd.
Exercise
A well-known rule used to find the square of an integer with decimal expansion (anan−1 … a1a0)10 and final digit a0 = 5 is to find the decimal expansion of the product (anan−1 … a1)10 [(anan−1 … a1)10 + 1], and append this with the digits (25)10. For instance, we see that the decimal expansion of (165)2 begins with 16 · 17= 272, so that (165)2 = 27,225. Show that this rule is valid.
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