Question

1. To prove 5+ √3 is an irrational number.
2. To prove √2 +√ 5 is an irrational number.
3. Insert five rational number or irrational number between √2 and 3.
4. If the HCF of 65 and 117 is of the form ( 65m-117) , then find m?
5. What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?
6. Find LCM of (23 x 3 x 5) and (24 x 5 x 7)
7. Find the simplest form of 148185​

Answer 1

1) Let us assume to the contrary that 5+√3 is rational.

That is we can find coprime a and (b≠0)

such that 5+√3 = a/b .

5+√3 = a/b => √3= a/b -5 =>√3=a-5b/b

Since a and b are integers, a-5b/b is rational.

So√3 is rational. But this contradicts the fact that√3 is irrational. So our assumption is wrong.

Therefore 5+√3 is irrational.

7) The simplest form of 148/185 is 4/5.

5) 13 is the largest no. which divides 70 and 125.