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2. Prove that p +2. Prove that p+Vg is irational, where p. q are primes. |
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Answer» First, we'll assume that √p + √q is rational, where p and q are distinct primes√p + √q = x, where x is rational Rational numbers are closed under multiplication, so if we square both sides, we still get rational numbers on both sides. (√p + √q)² = x²p + 2√(pq) + q = x²2√(pq) = x² - p - q √(pq) = (x² - p - q) / 2 Now x, x², p, q and 2 are all rational, and rational numbers are closed under subtraction and division. So (x² - p - q) / 2 is rational. But since p and q are both primes, then pq is not a perfect square and therefore √(pq) is not rational. But this is a contradiction. Original assumption must be wrong. So √p + √q is irrational, where p and q are distinct primes |
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