Prove the following statement by contraposition. Let x be an integer. If x2 + x + 1 is even, then x is odd. Make sure that your proof makes appropriate use of Definitions 1.5 and 1.6. Let x be an integer. Suppose that x is even. By Definition 1.5, x = for some integer k. Then x2 + x + 1 = ( )k + 1 = ( (2k² + k) + 1 is ---Select---, by Definition 1.6.

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Problems on numbers

Number theory is one of the most appealing topics in mathematics. Given one or more numeric values, a numeric algorithm performs some computation. The use of number theory in computing is important for applications like in the case of cryptographic algorithms. Number theory is also used in various applications such as calculating hash functions or in information security. Euclid’s algorithm is one of the oldest number-theoretic algorithms which is used to calculate the greatest common divisor of two numbers.