For how many different positive integers n does...?

Question

For how many different positive integers n does (sq. root of n) differ from (sq. root of 100) by less than 1?

Answer 1

In math terms we want to solve |sqrt(n) - 10| < 1 for positive integers n. We can rewrite this as

-1 < sqrt(n) - 10 < 1

                                  --> 9 < sqrt(n) < 11

                                  --> 81 < n < 121

which implies there are 120 - 81 = 39 such integers.

Answer 2

It would be all the integers from 82 to 120, or 39 in all.  

Answer 3

For how many different positive integers n does (sq. root of n) differ from (sq. root of 100) by less than 1?

gfxboy21 here it is:

√n + > 1 = √100

√n + > 1 = 10

√n  = 10 -> 1

√n ≈ 9

n ≈ 81 ← Ans

hope this helps