Base 2018

Number Theory Level 2

Suppose that % , # , @ and £ are symbols used to represent digits in base 2018 with the following properties:

% - # = @
% + # = @ + £
%# - #% = £@

What are the last 4 digits of the decimal representation of %£ - @# ?

Note: Remember that the symbols are digits, and therefore %# should not be read as % * # , in the same way that 12 is not read as 1 * 2

The answer is 7576.

1 solution

Stephen Mellor
Jan 6, 2018

If we take the information from the question and put it into an algebraic format we get,

% - # = @ (1)

% + # = @ + £ (2)

2018% + # - 2018# - % = 2018£ + @

which rearranged and factorised gives us,

2017(% - #) = 2018£ + @ (3)

Substituting $(1)$ into $(3)$ ,

2017@ = 2018£ + @

2016@ = 2018£

1008@ = 1009£

As all symbols must be between $1$ and $2017$ as they are in base $2018$ . Also, $1008$ and $1009$ are coprime, meaning the only possibility is $£ = 1008$ and $@ = 1009$ .

$(1) + (2)$ ,

2% = 2@ + £

Meaning that we can substitute in, and find that the values are $\% = 1513$ and $# = 504$ .

Placing these values into the final expression gives us a decimal value of $1017576$ , meaning the answer is $\boxed{7576}$ .

Base 2018

The answer is 7576.

1 solution

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