Description

Numerous are the moments I as a programmer have been frustrated by the tedious conversions between decimal numbers and hexadecimal ones. Why have we chosen 10 as a base in our everyday numerical presentations, when 16 seems so practically appealing? Obviously because everyone is not the computer geek I am. Maybe some day the world will fully realise the benefits of the hexadecimal system. In the meantime I have to learn to master the base conversions since most of the time numbers do not resemble one another in different bases.

Sometimes peculiar relationships emerge among the different base representations of numbers though. For instance, I noticed just the other day that 1040₁₀ * 4 = 1040₁₆, i.e. (1*10³+0*10²+4*10¹+0*10⁰)*4=(1*16³+0*16²+4*16¹+0*16⁰). It made me wonder how often this is the case, that is, the digits of a number in one base, are exactly the same as the digits of a multiple of the number in another base. Formally, let B1 < B2 be positive integers, and a0,a1,…,ak be integers in [0…B1-1]. For which ai’s is there a positive integer c such that

统计

递交统计

Base equality

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