Question

Discrete Math: Prove that there can be no perfect square between 25 and 36, i.e. there is no integer n so that 25 < n2 < 36. Prove this by directly proving the negation. Your proof must only use integers, inequalities and elementary logic. You may use that inequalities are prsered by adding a number on both sides, or by multiplying both sides by a positive number. You cannot use the square root function. Do not write a proof by contradiction. Please type answers so I can read them, and correctly learn how to perform these to improve my studies Thank you!

Answer 1

The only integers between 25 and 36 are 26, 27, 28, 29, 30, 31, 32, 33, 34, and 35

A number is a perfect square if and only if all the exponents appearing in its prime factorization are even for example $16=2^4\text{ and }36=2^2\cdot 3^2$

26 is not a perfect square as $26=2\times 13$

27 is not a perfect square as $27=3^3$

28 is not a perfect square as $28=2^2\times 7$

29 is not a perfect square as it is a prime

$30=2\times 3\times 5$ is also not a perfect square

31 is not a perfect square as it is a prime

32 is not a perfect square as $32=2^5$

33 is not a perfect square as $33=3\times 11$

34 is not a perfect square as $34=2\times 17$

35 is not a perfect square as $35=5\times 7$

Thus, there doesn't exist a perfect square between 25 and 36

$\blacksquare$

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