Hexadecimals without Letters

Number Theory Level 5

Let $S$ be the set of all real numbers $r < 1000$ whose hexadecimal (base 16) expansions do not contain any letters (i.e. they only have digits in the set $\{0, 1, 2, ... 9\}$ ).

Find the least upper bound of $S$ .

Details:

For example, $1.45 \in S$ because its hexadecimal expansion is $1.7\overline{3}_{16}$ , which only consists of the digits $1,3,7$ . However $240 \not \in S$ because its hexadecimal expansion is $f0_{16}$ which contains an $"f"$ .

Please write the answer in base 10, not 16.

Image credit: Wikipedia

The answer is 921.6.

1 solution

Bhaskar Sukulbrahman
Feb 28, 2015

${(1000)}_{10} = {(3E8)}_{16}$

So the largest integer, less than 1000 having no letters in base 16, is

${(399)}_{16} = {(921)}_{10}$

Now for largest fractional part:

${(0.9999....)}_{16} = {(0.6)}_{10}$

So the required number in base 10 is $\boxed{{(921.6)}_{10}} = {(399.9999...)}_{16}$ .

Hexadecimals without Letters

Image credit: Wikipedia

The answer is 921.6.

1 solution

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