2005 iTest Problems/Problem 17

Problem

On the $2004$ iTest, we defined an optimus prime to be any prime number whose digits sum to a prime number. (For example, $83$ is an optimus prime, because it is a prime number and its digits sum to $11$ , which is also a prime number.) Given that you select a prime number under $100$ , find the probability that is it not an optimus prime.

Solution 1

Consider the set of primes less than $100$ : $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$ . There are $25$ primes in this set. Since the last digit of a multi-digit prime number is odd, and the sum of two odd numbers is even, we can immediately disregard all primes with an odd tens digit. In addition, all the one-digit primes have a prime digit sum. We can consider the set of primes with a nonzero even tens digit and find their sums: $5, 11, 5, 7, 11, 7, 13, 11, 17$ . All of these sums are prime, so this case contributes $9$ primes. We can now find the desired probability: $1-\frac{9+4\text{(for the one digit primes)}}{25}=\boxed{\frac{12}{25}}$ .

2005 iTest (Problems, Answer Key)
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2005 iTest Problems/Problem 17

Problem

Solution 1

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