Question

prove that √10 is irrational number
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Answer 1

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Let$\sqrt{10}$ be a rational number.

Therefore

$\sqrt{10}$=$\frac{p}{q}$

or,$\sqrt{10}$ ×q =p

or,10q^2=p^2 (squaring both sides)

Therefore, 10 is a factor of q^2

And 10q^2=p^2............1

Therefore, 10 is also a factor of p^2

Therefore 10 is a factor of p

Therefore let p be equal to 10m(m greater than 0)

p=10m

or,p^2=100m^2.............2

By 1 and 2 we get,

100m^2=10q^2

or q^2=10m^2

Therefore 10 is a factor of m^2 and also q

Now we get p and q have common factor 10

But we know that in a rational number HCF(p,q)=1

Therefore$\sqrt{10}$is not a rational number.

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