2026 AIME I 第 4 题

Add $1$ to both sides: $a + b + ab + 1 = (a+1)(b+1)$

So $n$ works exactly when $n + 1$ is a product of two distinct factors, each at least $2;$ remove the primes and the squares of primes from $2$ through $101$

Since $a + b + ab = (a+1)(b+1) - 1,$ an integer $n$ is representable exactly when $n + 1 = xy$ for distinct integers $x = a + 1$ and $y = b + 1$ that are each at least $2.$ So we count integers $n + 1$ in $\{2, 3, \ldots, 101\}$ that admit such a factorization.

A prime has no factorization into two factors that are both at least $2,$ and the square of a prime $p^2$ factors that way only as $p \cdot p,$ which is not allowed. Every other composite $M$ works: if $p$ is its smallest prime factor, then $M = p \cdot \frac{M}{p}$ with $\frac{M}{p} \gt p$ since $M \gt p^2.$ In $\{2, \ldots, 101\}$ there are $26$ primes (the $25$ primes below $100,$ together with $101$) and $4$ prime squares ($4,$ $9,$ $25,$ $49$).

The count is $100 - 26 - 4 = 70.$