Problem 18 · 2009 Math Kangaroo
Stretch
Number Theory
perfect-squarecareful-counting
The difference between \(\sqrt{n}\) and 10 is less than 1. How many whole numbers n have this property?
Show answer
Answer: C — 39
Show hints
Hint 1 of 2
'√n differs from 10 by less than 1' means 9 < √n < 11.
Still stuck? Show hint 2 →
Hint 2 of 2
Square the bounds to get a range for n, then count the whole numbers strictly inside.
Show solution
Approach: square the inequality and count integers
- |√n − 10| < 1 means 9 < √n < 11.
- Squaring gives 81 < n < 121.
- The whole numbers strictly between are 82, 83, …, 120, which is 120 − 82 + 1 = 39.
Mark:
· log in to save