# On the definition of composite integers since n 2 2 n

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11. Using the method of proof by division into cases, prove the following statements:(a) Product of any two consecutive integers is even.
(b) For every integern,n2+ 3n-7 is odd.
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12. Use the method of exhaustion to prove the following statements:For every prime numberp <18, p-9 is not divisible by 10.pcan be one of the following numbers: 2, 3, 5, 7, 11, 13, and 17. So,p-9 can be oneof these numbers: -7, -6, -4, -2, 2, 4, and 8. Since none of these integers are divisibleby 10, the proof is complete.13. Prove the following statements by the method of indirect proof (proof by contradiction):(a)r, s, kZ,5r+ 15s-7
(b) sum of a rational and an irrational number is irrational.The formal representation of the given statement is:xr(x+r)/Q, wherexR-Q, rQBy contradiction, we assume that there is an irrational numberx /Qand arational numberrQsuch thatr+xQ.
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Chapter 3: Sets
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