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?
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Submitted by Federico Kereki
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Rating: 3.8333 (6 votes)
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Solution:
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200. You have to change its last number (or else it will be obviously not a prime) but 201, 203, 205, 207, and 209 are all composite. For any smaller number, you can change the last digit, and get 2,11, 23, 31, 41, 53, 61, 71, 83, 97, 101, 113, 127, 131, 149, 151, 163, 173, 181, or 191.
200+2310n gives an infinite family, because changing the last digit to 1 or 7 gives a number divisible by 3; to 3, a number divisible by 7; to 9, a number divisible by 11.
The proposed solution (about numbers N!+10, for N>18) is a nice alternative. |
