Find all 3 digit numbers such that by adding the first digit, the
square of the second and the cube of the third you get the original
number, for example 1 + 3^2 + 5^3 = 135.
If a number N is expressed in binary by using only 'ones,' what can
you say about its square (in binary)?
We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.
All the results can be checked by calculation. For an
explanation and proof it is most efficient to use a general method
that works for all bases. Moreover algebra reveals the structure
that is obscured when using particular numbers to check special
cases. Writing $b$ for the base then each expression can be written
in terms of $b$ and, by collecting like terms in the expression, or
equating the powers of $b$, it can be shown that each expression
holds. All that is required is some care over the algebraic