Let 5 be a rational number. then it must be in form of q p where, q =0 ( p and q are co-prime) 5 = q p 5 ×q=p Suaring on both sides, 5q 2 =p 2 --------------(1) p 2 is divisible by 5. So, p is divisible by 5. p=5c Suaring on both sides, p 2 =25c 2 --------------(2) Put p 2 in eqn.(1) 5q 2 =25(c) 2 q 2 =5c 2 So, q is divisible by 5. . Thus p and q have a common factor of 5. So, there is a contradiction as per our assumption. We have assumed p and q are co-prime but here they a common factor of 5. The above statement contradicts our assumption. Therefore, 5 is an irrational number.
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Two integers p and q are said to be r coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1
So according to this they both have only one common factor that is 1 so HCF =1
Correct answer will be Option D
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