What is the maximum number of 3-digit conscequetive integers that have atleast one odd digit?

Mar 8, 2016

997, 998 and 999.

Explanation:

If the numbers have at least one odd digit, in order to get the highest numbers let's chose 9 as the first digit. There is no restriction on the other digits, so the integers can be 997, 998 and 999.

Or you wanted to say at THE MOST one odd digit.

So let's chose 9 again. The other digits can not be odd. Since in three consecutive numbers, at least one must be odd, we cannot have three consecutive numbers in which 9 is the first digit.

So, we have to reduce the first digit to 8. If the second digit is 9, we can't have three consecutive numbers only with even numbers,unless the last of these numbers i 890, and the others are 889 and 888.

Mar 9, 2016

$111$

Explanation:

If I am interpreting the question correctly, it is asking for the length of the longest sequence of consecutive $3$-digit integers such that each integer contains at least one odd digit.

Any such sequence would necessarily include either $100 - 199$, $300 - 399$, $500 - 599$, $700 - 799$, or $900 - 999$.

We can discard $100 = 199$ as for any other sequence we gain additional values by subtracting from the lower end, whereas for $100$ we would go into $2$-digit integers, which are not allowed.

As adding $1$ to any of $399 , 599 , 799 , 999$ generates either an integer with no odd digits or with more than $3$ digits, one of those will be the greatest integer in the sequence. As there is no benefit to choosing one over another, we can choose one at random, say, $399$.

Counting down, as all of the $300$s have the first digit as odd, we only need to pay attention when we enter the $200$s. As we count down, all of the $290$s have the second digit as odd, and $289$ has the third digit as odd. Beyond that, we hit $288$ which would break the sequence. Similarly, if we tried with any other starting point, we would find that the longest sequence we could generate would be one of

$289 - 399$, $489 - 599$, $689 - 799$, or $889 - 999$.

each of which has a length of $111$.