Is 42 a rational number?

Question

eyakhiwabhn · Accepted Answer

First, calculate 42 as the sum of its prime factors.42=2×3×7There are no perfect squares and all the factors are prime numbers.2,3and7 are all irrational numbers.42 cannot be rational.

Jesse Freeman · Accepted Answer

Here is a diagram of a proof for what 42 is irrational:Suppose 42=pq where p&amp;gt;q&amp;gt;0 is the smallest pair of positive integers whose quotient is 42Therefore:p2=42q242=2⋅3⋅7So p2 is divisible by 2, 3 and 7.But if p2 is divisible by all of these primes, then p is divisible by all of them.Then:p=2⋅3⋅7k=42k for some positive integer k.So we have:42q2=p2=(42k)2=42⋅42k2By dividing the two ends, 42k2 we find:q2k2=42and hence:qk=42Now p&amp;gt;q&amp;gt;k&amp;gt;0, so q&amp;gt;k is a smaller pair of positive integers with quotient 42 contradicting our assertion that p&amp;gt;q is the smallest such pair.The smallest pair of positive integers p, q cannot therefore be pq=42So 42 is not expressible as pq for any pair of integers.In other words 42 is irrational.

Is sqrt(42) a rational number?

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