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Problem 2: (i) Let a be an integer. Prove that 2|a if and only if 2|a3....

Problem 2: (i) Let a be an integer. Prove that 2|a if and only if 2|a3.

(ii) Prove that 3√2 (cube root) is irrational.

Problem 3: Let p and q be prime numbers.

(i) Prove by contradiction that if p+q is prime, then p = 2 or q = 2

(ii) Prove using the method of subsection 2.2.3 in our book that if

p+q is prime, then p = 2 or q = 2

Proposition 2.2.3. For all n ∈ Z, if n2 is odd, then n is odd.

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Answer #1

a) consider fan=r7-2=0. Suppose e Paepe sot f(f/4)=0 Them by Rational root theorem, pl-2. and q11. possibilities for p are 1,-1,2,-2. - and that for & are 1 But if(1) = 4 f(-1)2 -3, f(2)= 6 F(-2)2 -20. Therefore fex)=0 Cannot have any rate 32 is real goot of f(n) zo. Hence 32 most be irrational.

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