Find the number of natural numbers $a$ that satisfy the inequality $5 < \sqrt{a} < 6$.

1. Problem Description

Find the number of natural numbers $a$ that satisfy the inequality $5 < \sqrt{a} < 6$.

2. Solution Steps

We are given the inequality $5 < \sqrt{a} < 6$. To find the possible values of $a$, we square all parts of the inequality.

Squaring all parts gives:

$5^2 < (\sqrt{a})^2 < 6^2$

$25 < a < 36$

Since $a$ is a natural number, we want to find the integers between 25 and 36, not including 25 and

3

6. The possible values for $a$ are 26, 27, 28, 29, 30, 31, 32, 33, 34,

3

5. To find the number of natural numbers that satisfy the inequality, we subtract the lower bound from the upper bound and subtract 1:

$36 - 25 - 1 = 11 - 1 = 10$

There are 10 natural numbers satisfying the inequality.

3. Final Answer

10