Question

The increasing sequence $3,$ $15,$ $24,$ $48,$ $\ldots$ consists of those positive multiples of $3$ that are one less than a perfect square. What is the remainder when the $1994$ th term of the sequence is divided by $1000?$

Answer

Answer: 63

Concepts:modular arithmetic counting integers in a range perfect square

Difficulty rating: 1640

Small Hint:

A number $k^2-1$ is divisible by $3$ exactly when $k$ is not divisible by $3$

Big Hint:

Index the eligible values of $k$ in pairs, then reduce the required square modulo $1000$

Solution:

The terms are $k^2-1$ for integers $k\geq2$ not divisible by $3.$ In each block of three consecutive $k$ ’s there are two eligible values. The $1994=2\cdot997$ th corresponds to $k=3(997)+1=2992.$ Since $2992\equiv992\equiv-8\pmod {1000},$ $\begin{aligned}k^2-1&\equiv(-8)^2-1\\&\equiv63\pmod {1000}.\end{aligned}$

1994 AIME Problem 1

Answer: 63

Difficulty rating: 1640

Small Hint:

Big Hint:

Solution:

Problem 1 in Other Years