Question

If $p$ and $q$ are primes and $x^2-px+q=0$ has distinct positive integral roots, then which of the following statements are true?

I. The difference of the roots is odd.

II. At least one root is prime.

III. $p^2-q$ is prime.

IV. $p+q$ is prime.

$\mathrm{I}$ only

$\mathrm{II}$ only

$\mathrm{II}$ and $\mathrm{III}$ only

$\mathrm{I},$ $\mathrm{II}$ and $\mathrm{IV}$ only

All are true.

Answer: E

Concepts:Vieta’s Formulas prime factor

Difficulty rating: 1830

Small Hint:

The product of the two positive integer roots is the prime $q$

Big Hint:

The roots must be $1,q,$ and their prime sum forces $q=2$

Solution:

The positive integer roots have product $q,$ so they are $1$ and $q.$ Their sum is $p=q+1,$ which is prime only when $q=2,$ giving $p=3$ and roots $1,2.$ Their difference is $1,$ one root is prime, $p^2-q=7,$ and $p+q=5;$ therefore all four statements hold.

Therefore, the correct answer is E.

1975 AMC 12 Problem 22

Answer: E

Difficulty rating: 1830

Small Hint:

Big Hint:

Solution:

Problem 22 in Other Years