Decimal to Binary Form

Number Theory Level 3

I am looking for a positive integer $X$ , which is smaller than 2000, such that: $(2018 + X)^2 - 1$ in decimal form gives no zeroes in binary form.

What is $X$ ?

480 60 960 1920 30 240 15 120

1 solution

Jordan Cahn
Oct 27, 2018

If a number has no zeros in binary, it must be of the form $2^n-1$ . Thus, $2018+X$ must be a power of $2$ . Since $X$ is positive, we are looking for a power of two greater than $2018$ . One candidate is $2048$ , in which case $X=30$ . The next candidate is $4096$ , which would mean that $X=2078$ , which is too large. Thus $X=\boxed{30}$ .

Decimal to Binary Form

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