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Difference between revisions of "1979 AHSME Problems/Problem 22"

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(The first solution is total nonsense!)
 
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==Solution==
 
==Solution==
Solution by e_power_pi_times_i
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The equation is equivalent to <math>m(m+1)(m+5) = 27n^3 + 27n^2 + 9n + 1</math>. Taking mod 3, we get
 
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<math>m(m+1)(m+2)=1 (\bmod 3)</math>. However, <math>m(m+1)(m+2)</math> is always divisible by <math>3</math> for any integer <math>m</math>. Thus, the answer is <math>\boxed{\textbf{(A)} 0}</math>
Notice that <math>m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1 = \frac{(3n+1)^3}{27}</math>. Then <math>27(m^3 + 6m^2 + 5m) = (3n+1)^3</math>, and <math>(3n+1)^3 | 27</math>. However, <math>(3n+1)^3</math> will never be divisible by <math>3</math>, nor <math>27</math>, so there are <math>\boxed{\textbf{(A)} 0 }</math> integer pairs of <math>(m, n)</math>.
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Solution by mickyboy789
  
 
== See also ==
 
== See also ==

Latest revision as of 09:26, 19 August 2021

Problem 22

Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1$.

$\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }3\qquad \textbf{(D) }9\qquad \textbf{(E) }\infty$

Solution

The equation is equivalent to $m(m+1)(m+5) = 27n^3 + 27n^2 + 9n + 1$. Taking mod 3, we get $m(m+1)(m+2)=1 (\bmod 3)$. However, $m(m+1)(m+2)$ is always divisible by $3$ for any integer $m$. Thus, the answer is $\boxed{\textbf{(A)} 0}$ Solution by mickyboy789

See also

1979 AHSME (ProblemsAnswer KeyResources)
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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