Difference between revisions of "2001 AMC 10 Problems/Problem 12"
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== Problem == | == Problem == | ||
| − | |||
Suppose that <math> n </math> is the product of three consecutive integers and that <math> n </math> is divisible by <math> 7 </math>. Which of the following is not necessarily a divisor of <math> n </math>? | Suppose that <math> n </math> is the product of three consecutive integers and that <math> n </math> is divisible by <math> 7 </math>. Which of the following is not necessarily a divisor of <math> n </math>? | ||
<math> \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 42 </math> | <math> \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 14 \qquad \textbf{(C)}\ 21 \qquad \textbf{(D)}\ 28 \qquad \textbf{(E)}\ 42 </math> | ||
| − | == Solution == | + | == Solutions == |
| + | === Solution 1 === | ||
Whenever <math> n </math> is the product of three consecutive integers, <math> n </math> is divisible by <math> 3! </math>, meaning it is divisible by <math> 6 </math>. | Whenever <math> n </math> is the product of three consecutive integers, <math> n </math> is divisible by <math> 3! </math>, meaning it is divisible by <math> 6 </math>. | ||
| Line 12: | Line 12: | ||
In our answer choices, the one that is not a factor of <math> 42 </math> is <math> \boxed{\textbf{(D)}\ 28} </math>. | In our answer choices, the one that is not a factor of <math> 42 </math> is <math> \boxed{\textbf{(D)}\ 28} </math>. | ||
| + | |||
| + | === Solution 2 === | ||
| + | |||
| + | We can look for counterexamples. For example, letting <math>n = 13 \cdot 14 \cdot 15</math>, we see that <math>n</math> is not divisible by 28, so <math>\boxed{\textbf{(D) }28}</math> is our answer. | ||
| + | |||
| + | == See Also == | ||
| + | |||
| + | {{AMC10 box|year=2001|num-b=11|num-a=13}} | ||
| + | {{MAA Notice}} | ||
Revision as of 10:45, 8 November 2021
Problem
Suppose that 
is the product of three consecutive integers and that is divisible by 
. Which of the following is not necessarily a divisor of ?
Solutions
Solution 1
Whenever is the product of three consecutive integers, is divisible by 
, meaning it is divisible by 
.
It also mentions that it is divisible by , so the number is definitely divisible by all the factors of 
.
In our answer choices, the one that is not a factor of is 
.
Solution 2
We can look for counterexamples. For example, letting 
, we see that is not divisible by 28, so 
is our answer.
See Also
| 2001 AMC 10 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 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 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
