Question

A calculator has a squaring key $\boxed{x^2}$ that replaces the number currently displayed with its square. For example, if the display reads $\boxed{\phantom{00}3}$ and $\boxed{x^2}$ is pressed, the display becomes $\boxed{\phantom{00}9}.$ If the display reads $\boxed{\phantom{00}2},$ how many times must $\boxed{x^2}$ be pressed to produce a displayed number greater than $500?$

$4$

$5$

$8$

$9$

$250$

Answer: A

Concepts:exponent

Difficulty rating: 1030

Small Hint:

Starting from $2,$ each press squares the number: $2,$ $4,$ $16,$ $256,$ $\ldots$

Big Hint:

Find the first term in that list that exceeds $500$

Solution:

Pressing the key repeatedly gives $2 \to 4 \to 16 \to 256 \to 65536.$

Since $256 \lt 500 \lt 65536,$ the display first exceeds $500$ on the fourth press.

Thus, the correct answer is A .

1987 AMC 8 Problem 19

Answer: A

Difficulty rating: 1030

Small Hint:

Big Hint:

Solution:

Problem 19 in Other Years