Problem 7 · Factor a difference of squares and count symmetric solutions · 5 points
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How many integers n make n² + 30n + 81 the square of an integer?
- 8
- 9
- 5
- 10
- 3
Show a hint
- There’s a square hiding in the quadratic. What happens if you compare it with the square you’re looking for?
- Rewrite the expression as (n + 15)² − 144, then factor the difference of two squares. Check the parity of the factors and both signs of n + 15.
Show the answer and solution
Answer: D — 10
Suppose n² + 30n + 81 = m². Then (n + 15 − m)(n + 15 + m) = 144. The two factors have the same parity, so they must both be even. Their positive factor pairs are (2, 72), (4, 36), (6, 24), (8, 18), and (12, 12). Each pair gives one value of n + 15 and its negative, hence two values of n, for 10 altogether. The tempting answer 5 counts only one sign of n + 15; the negative sign also gives valid integers.