Prove any prime number is irrational, Mathematics

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1. Show that there do not exist integers x and y for which 110x + 315y = 12.

2. If a and b are odd integers, prove that a² +b² is divisible by 2 but is NOT divisible by 4. Hint:

Any odd number can be written 2k + 1 for some integer k.

3. Prove that for any prime number p, √ p is irrational. Hint: Follow the same method that was used to prove √ 2 is irrational, by first supposing √ p can be expressed as a=b.

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