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4. (a) Let n = (dkdk−1 . . . d0)b. Prove that (b + 1)|n if...

4. (a) Let n = (dkdk−1 . . . d0)b. Prove that (b + 1)|n if and only if (b + 1)|d0 − d1 + d2 − · · · + (−1)kdk.

(b) A number is a palindrome if reversing the sequence of its digits gives the same number. For example, 12321 and 456654 are palindromes. Use part (a) with b = 10 to prove that every palindrome with an even number of digits is divisible by 11.

(c) Which palindromes with an odd number of digits are divisible by 11? Why?

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