Ones on Each Side

Number Theory Level pending

Let $x$ be a five digit number. (This implies that $x$ is a positive integer, with $9999 < x < 100000$ ) If the number $1$ was written in front of $x$ , then it would be one third of if the number $1$ was written at the end of $x$ . (For example, writing the number $1$ in front of $576$ would be $1576$ , and writing it at the end would be $5761$ ).

What is $x$ ?

The answer is 42857.

1 solution

Ray Chou
Jul 13, 2015

Though one may be tempted to individually solve for all of the digits (which can lead to a correct solution), there is a much more elegant method.

Because $x$ is a five digit number, note that $100000 + x$ is adding $1$ to the front of the number, and $10x + 1$ adds a $1$ to the end of the number.

So we have that $3(100000+x) = 10x + 1$

$299999 = 7x$

$x = \boxed{42857}$

Remark: Note that the decimal expansion of $\frac{1}{7}$ is $0.142857...$ , and that the other multiples of this fraction less than $1$ are the same numbers repeated in the same order, just starting on a different digit.

Ones on Each Side

The answer is 42857.

1 solution

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