1979 AHSME Problems/Problem 22

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

Solution by e_power_pi_times_i

Notice that $m^3 + 6m^2 + 5m = 27n^3 + 27n^2 + 9n + 1 = \frac{(3n+1)^3}{27}$ . Then $27(m^3 + 6m^2 + 5m) = (3n+1)^3$ , and $(3n+1)^3 | 27$ . However, $(3n+1)^3$ will never be divisible by $3$ , nor $27$ , so there are $\boxed{\textbf{(A)} 0 }$ integer pairs of $(m, n)$ .

Solution 2: The equation is equivalent to $m(m+1)(m+5) = 27n^3 + 27n^2 + 9n + 1$ . Taking mod 3, we get $m(m+1)(m+2)=1 (mod 3)$ or $0=1 (mod 3)$ which is a contradiction. Thus, the answer is $(A) 0$

1979 AHSME (Problems • Answer Key • Resources)
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
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1979 AHSME Problems/Problem 22

Problem 22

Solution

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