Largest three digit number comprising of only 1's and 0's that when divided by 23 gives a remainder of 18

Question

So I had a Year 11/12 Maths Competition today and one of the last questions on the competition paper was this:

"A certain number is comprised of only 1's and 0's. When divided by 23 it has a remainder of 18. What is the highest amount of "1's" in a number that fits this description?"

to my understanding, this means the number is Base 10 and not binary

this question was not multiple choice, you were given 3 digits to write into (keep in mind, the first number can be zero)

The test was also done without a calculator so if you can provide steps without the use of a calculator and what we can calculate on rough paper, that would be great. Thanks

Answer 1

If you’re limited to three-digit numbers (possibly with leading zeroes), you might as well do it by brute force: start with $18$ and keep adding $23$. You very quickly get

$$18,41,64,87,\underline{110},133,156,179\;,$$

and the next number is too large. The only three-digit number that fits the description is $110$, and it has $2$ ones.

If you’re not limited to three digits, then there is no largest possible number of ones, as others have already pointed out in the comments.