1975 AMC 12 Problem 22

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.

The product of the two positive integer roots is the prime

q

The roots must be

1,q,

and their prime sum forces

q=2

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.

Problem 22 in other years