Difference between revisions of "1975 AHSME Problems/Problem 22"
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+ | ==Problem== | ||
+ | If <math>p</math> and <math>q</math> are primes and <math>x^2-px+q=0</math> has distinct positive integral roots, then which of the following statements are true? | ||
+ | |||
+ | <math> | ||
+ | I.\ \text{The difference of the roots is odd.} \\ | ||
+ | II.\ \text{At least one root is prime.} \\ | ||
+ | III.\ p^2-q\ \text{is prime}. \\ | ||
+ | IV.\ p+q\ \text{is prime} | ||
+ | </math> | ||
+ | |||
+ | <math> | ||
+ | \\ | ||
+ | \textbf{(A)}\ I\ \text{only} \qquad | ||
+ | \textbf{(B)}\ II\ \text{only} \qquad | ||
+ | \textbf{(C)}\ II\ \text{and}\ III\ \text{only} \\ | ||
+ | \textbf{(D)}\ I, II, \text{and}\ IV\ \text{only}\ \qquad | ||
+ | \textbf{(E)}\ \text{All are true.} | ||
+ | </math> | ||
+ | |||
+ | ==Solution== | ||
+ | |||
Since the roots are both positive integers, we can say that <math>x^2-px+q=(x-1)(x-q)</math> since <math>q</math> only has <math>2</math> divisors. Thus, the roots are <math>1</math> and <math>q</math> and <math>p=q+1</math>. The only two primes which differ by <math>1</math> are <math>2,3</math> so <math>p=3</math> and <math>q=2</math>. | Since the roots are both positive integers, we can say that <math>x^2-px+q=(x-1)(x-q)</math> since <math>q</math> only has <math>2</math> divisors. Thus, the roots are <math>1</math> and <math>q</math> and <math>p=q+1</math>. The only two primes which differ by <math>1</math> are <math>2,3</math> so <math>p=3</math> and <math>q=2</math>. | ||
<math>I</math> is true because <math>3-2=1</math>. | <math>I</math> is true because <math>3-2=1</math>. | ||
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Thus, the answer is <math>\textbf{(E)}</math>. | Thus, the answer is <math>\textbf{(E)}</math>. | ||
-brainiacmaniac31 | -brainiacmaniac31 | ||
+ | |||
+ | ==See Also== | ||
+ | {{AHSME box|year=1975|num-b=21|num-a=23}} | ||
+ | {{MAA Notice}} |
Latest revision as of 17:48, 19 January 2021
Problem
If and are primes and has distinct positive integral roots, then which of the following statements are true?
Solution
Since the roots are both positive integers, we can say that since only has divisors. Thus, the roots are and and . The only two primes which differ by are so and . is true because . is true because one of the roots is which is prime. is true because is prime. is true because is prime. Thus, the answer is . -brainiacmaniac31
See Also
1975 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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