The crisp new dollar bill that I have just taken from my wallet bears the serial number 24598176. It’s easy to tell that the number is exactly divisible by 2 but not by 5. Is it divisible by 3? by 4?

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We give a characterization of functions of the form f(n) = ν(n - E), where ν(-) denotes the exponent of 2, and E is a 2-adic integer. We show that it applies to the restriction to even or odd integers ...