Difference between revisions of "1994 AHSME Problems/Problem 21"

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<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 </math>
 
<math> \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 </math>
 
==Solution==
 
==Solution==
Since the sum of the digits of <math>N</math> is <math>4</math> and none of the digits are <math>0</math>, <math>N</math>'s digits must be the elements of the sets <math>\{1,1,1,1\},\{1,1,2\}</math>, <math>{2,2}</math>, <math>\{1,3\}</math>, or <math>\{4\}</math>.  
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Since the sum of the digits of <math>N</math> is <math>4</math> and none of the digits are <math>0</math>, <math>N</math>'s digits must the elements of one of the sets <math>\{1,1,1,1\},\{1,1,2\}</math>, <math>\{2,2\}</math>, <math>\{1,3\}</math>, or <math>\{4\}</math>.  
  
 
In the first case, <math>N = 1111 = 101 \cdot 11</math> so this is a counter example.   
 
In the first case, <math>N = 1111 = 101 \cdot 11</math> so this is a counter example.   

Latest revision as of 14:11, 28 May 2021

Problem

Find the number of counter examples to the statement: \[``\text{If  N is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then N is prime}."\] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Solution

Since the sum of the digits of $N$ is $4$ and none of the digits are $0$, $N$'s digits must the elements of one of the sets $\{1,1,1,1\},\{1,1,2\}$, $\{2,2\}$, $\{1,3\}$, or $\{4\}$.

In the first case, $N = 1111 = 101 \cdot 11$ so this is a counter example.

In the second case, $N=112$ is excluded for being even. With $N=121=11^2$ we have a counterexample. We can check $N=211$ by trial division, and verify it is indeed prime.

In the third case, $N=22$ is excluded for being even.

In the fourth case, both $N=13$ and $N=31$ are prime.

In the last case $N=4$ is excluded for being even.

This gives two counterexamples and the answer is $\fbox{C}$

See Also

1994 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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