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Difference between revisions of "1951 AHSME Problems/Problem 18"

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== Solution 2==
 
== Solution 2==
 
Factoring <math>21x^2+ax+21</math> by grouping, we need to find some <math>b,c</math> such that <math>b\cdot c = 21^2</math>, and that <math>b+c=a</math>.
 
Factoring <math>21x^2+ax+21</math> by grouping, we need to find some <math>b,c</math> such that <math>b\cdot c = 21^2</math>, and that <math>b+c=a</math>.
−
Since <math>21^2\equiv 1 (mod2)</math>, <math>b\land c \equiv 1 (mod 2)</math>, and <math>b+c \equiv 0 (mod 2)</math>. So <math>a</math> must be even. <math>a</math> cannot be any even, since <math>21^2</math> only has 4 odd factors, so the answer is <math> \textbf{(D)}\ \text{some even number}</math>
+
Since <math>21^2\equiv 1 (mod\!2)</math>, <math>b\land c \equiv 1 (mod 2)</math>, and <math>b+c \equiv 0 (mod\!2)</math>. So <math>a</math> must be even. <math>a</math> cannot be any even, since <math>21^2</math> only has 4 odd factors, so the answer is <math> \textbf{(D)}\ \text{some even number}</math>
  
 
== See Also ==
 
== See Also ==

Revision as of 10:55, 19 March 2017

Problem

The expression $21x^2 +ax +21$ is to be factored into two linear prime binomial factors with integer coefficients. This can be done if $a$ is:

$\textbf{(A)}\ \text{any odd number} \qquad\textbf{(B)}\ \text{some odd number} \qquad\textbf{(C)}\ \text{any even number}$ $\textbf{(D)}\ \text{some even number} \qquad\textbf{(E)}\ \text{zero}$

Solution

We can factor $21x^2 + ax + 21$ as $(7x+3)(3x+7)$, which expands to $21x^2+42x+21$. So the answer is $\textbf{(D)}\ \text{some even number}$

Solution 2

Factoring $21x^2+ax+21$ by grouping, we need to find some $b,c$ such that $b\cdot c = 21^2$, and that $b+c=a$. Since $21^2\equiv 1 (mod\!2)$, $b\land c \equiv 1 (mod 2)$, and $b+c \equiv 0 (mod\!2)$. So $a$ must be even. $a$ cannot be any even, since $21^2$ only has 4 odd factors, so the answer is $\textbf{(D)}\ \text{some even number}$

See Also

1951 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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All AHSME Problems and Solutions

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