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 (except for 11, which has digit sum $2$ ). 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: $1-\frac{9+1+4\text{(for the one digit primes)}}{25}=\boxed{\frac{11}{25}}$ .

2005 iTest (Problems, Answer Key)
Preceded by: Problem 16	Followed by: Problem 18
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2005 iTest Problems/Problem 17

Problem

Solution 1

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