Difference between revisions of "1959 AHSME Problems/Problem 42"
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| − | == Problem | + | == Problem == |
| − | Given three positive integers <math>a,b,</math> and <math>c</math>. Their greatest common divisor is <math>D</math>; their least common multiple is <math> | + | Given three positive integers <math>a,b,</math> and <math>c</math>. Their greatest common divisor is <math>D</math>; their least common multiple is <math>M</math>. |
Then, which two of the following statements are true? | Then, which two of the following statements are true? | ||
<math>\text{(1)}\ \text{the product MD cannot be less than abc} \qquad \\ | <math>\text{(1)}\ \text{the product MD cannot be less than abc} \qquad \\ | ||
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\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs} \text{ (This means: no two have a common factor greater than 1.)}</math> | \text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs} \text{ (This means: no two have a common factor greater than 1.)}</math> | ||
<math>\textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 </math> | <math>\textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 </math> | ||
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== Solution == | == Solution == | ||
Because <math>1\times2\times4>1\times4</math>, 1 is false. Because <math>1\times1\times1=1\times1</math>, 3 is false. It follows that the answer is <math>\boxed{\textbf{E}}</math>. | Because <math>1\times2\times4>1\times4</math>, 1 is false. Because <math>1\times1\times1=1\times1</math>, 3 is false. It follows that the answer is <math>\boxed{\textbf{E}}</math>. | ||
Latest revision as of 10:52, 22 July 2024
Problem
Given three positive integers 
and 
. Their greatest common divisor is 
; their least common multiple is 
.
Then, which two of the following statements are true?
Solution
Because 
, 1 is false. Because 
, 3 is false. It follows that the answer is 
.
See also
| 1959 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 41 |
Followed by Problem 43 | |
| 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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
