12 can be made into a prime by changing the 2 to 3; likewise, 63 can be changed into a prime by changing the 6 to 7.

What is the smallest number that cannot changed into a prime by changing a single digit?

Are there infinitely many such numbers?

The following infinite sequence will work:

200, 23300, 46400, ...

where the difference is 23100

The sum of the digits of each of these is 1 less than a multiple of 3 so that changing the last digit to a 1 or a 7 will make the number a multiple of 3.

203 is a multiple of 7 as is the difference, 23100, so whenever the last digit of any of these is changed to 3, the number is a multiple of 7.

209 is a multiple of 11 and so is 23100, so in all the cases where the last digit is changed to 9 the result is a multiple of 11.

The even last digits are again multiples of 2 and a last digit of 5 results in a multiple of 5.  Changing any digit but the last leaves a multiple of 10.

There are of course numbers that fit the pattern that are not on this sequence, but this proves the infinite number of such numbers.

Posted by Charlie on 2004-04-29 11:25:55