Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 year ago

# Proving √5321 − 1 is an irrational number?

I'm stuck on a textbook problem where I'm asked to prove that √5321 − 1 is an irrational number by contradiction and prime factorization. I don't really know where to begin, all I know is I start by equating it to a/b. Can someone walk me through this step by step?

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Lets suppose that √5321 − 1 is rational. Since -1 is rational, then, this would imply that √5321 is also rational as the sum of a rational number and an irrational number is also irrational. Let it be m/n, where m and n are integers with no common factors other than 1. That is, m/n is an irreducible fraction. Then, m²/n² = 5321, or, m²= 5321n². Since m and n are coprime and m² = 5321n², this implies that 5321 is a factor of m². Therefore, there is some integer k such that m² = 5321k = 5321n². Therefore, we may write n² = k. Thus n divides k. But if n divides k, which divides m, then n divides m. This is a contradiction since we assumed that m and n are coprime. Thus the assumption that √5321 was rational was in error. Therefore, √5321 must be irrational. Thus, √5321 - 1 is irrational.

An alternate approach is to note that 5321 = 17 × 313, both of which are prime. Thus, 5321 is not a perfect square and its root is not rational. Thus its root minus 1 is also irrational.

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• sqr(5321-1)

=

sqr(5320)

=

2sqr(5*2*133)

=

2sqr(2*5*7*19)

2,5,7 & 19 are primes

their product must not

be a square number. Thus

sqr(5320) is an irrational number.

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