Question

Grogg typed the following $1000$ expressions into his calculator, one by one: \[\sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \dots, \sqrt{999}, \sqrt{1000}. \]how many times will grogg's result be an integer?

Answer 1

Since 30² = 900 and 40² = 1600, between 30 and 40 of the numbers Grogg enters will reduce to integers. Now just find which count is correct. We don't have to look far:

31² = 961

32² = 1024

so there 31 of the outputs of √1, √2, √3, ..., √1000 are integers.