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?
Since there are prime numbers in each group of ten (1s, 10s, 20s) up until the 190s it is always possible to create a prime by changing the ones digit.
Taking the number 200, one can't change the hundreds or tens digit to make a prime because it would end in 0 and automatically be even. Since there is no prime number between 200 and 209 the ones digit cannot be changed either so 200 is the smallest number.
Next we know that there might be others in this group of 10 but since 101, 103, 107 and 109 are primes we rule those out. since 202, 204, 205, 208 are all divisible by either 2 or 5 and the units digit cannot be changed to make a prime number these also follow the rule.
The next group of 10 that there are no prime numbers is the 320 group. we know again that any even numbers or numbers that end in 5 or 0 cannot be changed into primes. 320, 322, 324, 325, 326, 328 and the others need to be checked. 321 - change to 421, 323 - 223, 327 - 127, 329 - 229.
For each group of 10 numbers that have no primes, there are atleast 6 that cannot be changed into primes (the numbers ending with 0,2,4,5,6,8).
Edited on April 29, 2004, 8:45 am
1st part
