Advertisements
Advertisements
प्रश्न
Prove that for any prime positive integer p, \[\sqrt{p}\]
is an irrational number.
उत्तर
Let us assume that `sqrtq` is rational .Then, there exist positive co primes a and b such that
`sqrtq=a/b`
`p= (a/b)^2`
`⇒ p = a^2/b^2`
`⇒ pb^2=a^2`
`⇒ pb^2=a^2`
`⇒ p|a^2`
`⇒ p|a`
`⇒ a= pc`for some positive intger c
`⇒ b^2p=a^2`
`⇒ b^2 p = p^2c^2(because a= pc)`
`⇒ p|b^2(since p|c^2p)`
`⇒ p|b`
`⇒ p|a and p|b`
This contradicts the fact that a and b are co priumes
Hence `sqrtp`is irrational
APPEARS IN
संबंधित प्रश्न
Using Euclid's division algorithm, find the H.C.F. of 196 and 38220
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?
144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?
Express each of the following as a rational number in its simplest form:
(i`) 0.bar (8)`
The LCM of two numbers is 1200, show that the HCF of these numbers cannot be 500. Why ?
Show that \[3 + \sqrt{2}\] is an irrational number.
Show that every positive odd integer is of the form (4q + 1) or (4q + 3), where q is some integer.
Find the largest number which on dividing 1251, 9377 and 15628 leave remainders 1, 2 and 3 respectively.
The largest number that will divide 398,436 and 542 leaving remainders 7,11 and 15 respectively is ______.
“The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.
