X, Y, and Z proofPro Problems > Math > Logic > Proofs > Indirect Proofs
X, Y, and Z proof
Prove by contradiction: for any integers x, y, and z with z > 1, x2 - yz2 ≠ z.
Hint: This is a more generalized version of the problem found here: X Squared Proof
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Let a and b be positive integers, with b not a perfect square. Show that the following expression is irrational, using an indirect proof.
Hint: Remember that an irrational number is a number which cannot be written as the quotient of two integers.
Hint: An irrational number is a number which cannot be written as the quotient of two relatively prime integers.
Prove by indirect method that if x and y are integers, then x2 - 9y ≠ 3.
For every positive rational number x, there exists a smaller positive rational number y. Prove this statement by indirect proof.
Use an indirect proof to show that there are infinitely many prime numbers.
Show that if θ is between 0º and 90º inclusive, then sin θ + cos θ ≥ 1. Use an indirect proof.
Create an indirect proof for the following statement: The harmonic mean of any two positive numbers lies inclusively between the two numbers.