1990 AHSME Problems/Problem 19

Problem

For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ $\underline{not}$ in lowest terms?

$\text{(A) } 0\quad \text{(B) } 86\quad \text{(C) } 90\quad \text{(D) } 104\quad \text{(E) } 105$

Solution

What we want to know is for how many $n$ is $\gcd(n^2+7, n+4) > 1.$ We start by setting $n+4 \equiv 0 \mod m$ for some arbitrary $m$ . This shows that $m$ evenly divides $n+4$ . Next we want to see under which conditions $m$ also divides $n^2 + 7$ . We know from the previous statement that $n \equiv -4 \mod m$ and thus $n^2 \equiv (-4)^2 \equiv 16 \mod m.$ Next we simply add $7$ to get $n^2 + 7 \equiv 23 \mod m.$ However, we also want $n^2 + 7 \equiv 0 \mod m$ which leads to $n^2 + 7\equiv 23 \equiv 0 \mod m$ from the previous statement. From that statement, we get that $m$ divides $23$ evenly. Since $23$ is prime and we're looking for a GCD greater than 1, $m$ must be $23$ . Going back to our original statement, we can set $n+4=23x$ for some positive integer x, and $n=23x-4.$ Finally, we must find the largest $x$ such that $23x-4<1990.$ This is a simple linear inequality for which the answer is $x=86$ , or $\fbox{B}$ .

1990 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 18	Followed by Problem 20
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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1990 AHSME Problems/Problem 19

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