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

1979 AMC 12 Problem 22

Attempt Problem 22 of the 1979 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1979 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

22.

Find the number of pairs

(m,n)

of integers which satisfy the equation

\begin{aligned} m^3+6m^2+5m &=27n^3+9n^2\\ &\quad+9n+1. \end{aligned}

0

1

3

9

infinitely many

Factor the left side as

m(m+1)(m+5)

Compare the two sides modulo

3

The left side is

\begin{gathered} m(m+1)(m+5)\\ =m(m+1)(m+2)\\ \quad+3m(m+1), \end{gathered}

which is divisible by

3

for every integer

m.

The right side is

3(9n^3+3n^2+3n)+1,

which is congruent to

1\pmod3.

Therefore no integer pair satisfies the equation.

Therefore, the correct answer is A.