Prove using only integers and inequalities that there is no

Prove using only integers and inequalities that there is no natural number whose square is 10. (Producing a decimal approximation of square 10, or arguing that squareroot 10 is between 2 and 3 is not proof. Just because one solution of x^2 = 10 is not an integer doesn\'t mean that there couldn\'t be some other that is an integer. Do not rework the method that is given in many textbooks for showing that squareroot s of non-perfect squares are irrational. Do not use number-theoretic concepts such as divisibility or relative primarily. Your solution must contain no references to squareroot s or quotients, or use algebraic facts concerning the solutions of quadratic equations.)

Solution

10<16

Hence,sqrt{10}<4

9<10

Hence, 3<sqrt{10}<4

But there is not natural number between 3 and 4

Hence there is no natural number who square is 10