2003 CEMC Pascal Problems/Problem 9

Problem

The largest prime number less than 30 that can be written as the sum of two primes is

$\text{ (A) }\ 29 \qquad\text{ (B) }\ 23 \qquad\text{ (C) }\ 19 \qquad\text{ (D) }\ 17 \qquad\text{ (E) }\ 13$

Solution 1

We can see that $2$ must be one of the primes included in the sum because $2$ is the only even prime number, and adding two odd numbers would result in an even number.

Listing all of the primes above $2$ that are below $30$ we have $3, 5, 7, 11, 13, 17, 19, 23, \text{and} 29$ .

$2 + 29 = 31$ , which goes above $30$ , so we can get rid of it.

Of the remaining primes, the only ones that are still prime when added to $2$ are:

$3, 5, 11, \text{and} 17$

The highest number here is $17$ , and $2 + 17 = \boxed {\textbf {(C) } 19}$

~anabel.disher

Solution 2

We can use the same logic as solution 1 to deduce that one of the prime numbers is $2$ . This means that we can just subtract $2$ from the answer choices and go from highest to lowest until we get a prime number:

$29 - 2 = 27$ , which is divisible by $3$ , so it is not prime

$23 - 1 = 21$ , which is divisible by $3$ , so it is not prime

$19 - 2 = 17$ , which is prime

Thus, $\boxed {\textbf {(C) } 19}$ is the answer

~anabel.disher

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2003 CEMC Pascal Problems/Problem 9

Problem

Solution 1

Solution 2