Real Numbers — Practice

Question 1

Find the HCF of 96 and 404 using Euclid's division algorithm.

Accepted Answer

HCF = 4. Steps: 404 = 96·4 + 20; 96 = 20·4 + 16; 20 = 16·1 + 4; 16 = 4·4 + 0. Last non-zero remainder = 4.

Question 2

Without doing the proof, list which of these are irrational: √7, √16, √(9/25), √20, 2π.

Accepted Answer

Irrational: √7, √20, 2π. Rational: √16 = 4, √(9/25) = 3/5.

Question 3

If HCF(72, 120) = 24, find LCM(72, 120) using the identity.

Accepted Answer

LCM = (72 × 120) / 24 = 8640 / 24 = 360.

Question 4

Without long division, decide whether 17/6250 has a terminating decimal expansion.

Accepted Answer

6250 = 2 · 5⁵. Form is 2ⁿ · 5ᵐ → YES, it terminates (= 0.00272).

Question 5

Two tankers contain 850 L and 680 L of petrol. Find the maximum capacity of a container that can measure both exactly.

Accepted Answer

We need HCF(850, 680). Prime factorisation: 850 = 2 · 5² · 17, 680 = 2³ · 5 · 17. HCF = 2 · 5 · 17 = 170 L.

Question 6

Prove that 5 − 2√3 is irrational, given √3 is irrational.

Accepted Answer

Suppose 5 − 2√3 = r is rational. Then √3 = (5 − r)/2. RHS is rational (rationals are closed under subtraction and division by 2), so √3 would be rational — contradiction. Hence 5 − 2√3 is irrational.

Question 7

The HCF of two numbers is 27 and their LCM is 162. If one number is 54, find the other.

Accepted Answer

Using HCF × LCM = product: 27 · 162 = 54 · x ⇒ x = (27 · 162) / 54 = 81.

Practice — Real Numbers

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