Real Numbers Important Questions & DPP with Solutions

State the Fundamental Theorem of Arithmetic.

Find HCF and LCM of 96 and 404 by prime factorisation method.

Prove that √2 is irrational.

The HCF of two numbers is 18 and their LCM is 2160. If one number is 360, find the other.

Check whether 6ⁿ can end with digit 0 for any natural number n.

Prove that √3 is irrational. Hence show that 2 + √3 is also irrational.

Explain Euclid's Division Lemma. Use it to find HCF(135, 225).

Without performing long division, state whether 17/8 will have terminating or non-terminating decimal expansion. Give reason.

Prove that 3 + 2√5 is irrational.

Find the LCM of 12, 15 and 21 using prime factorisation.

If HCF(a, b) = 6 and a × b = 216, find LCM(a, b).

Show that any positive odd integer is of the form 6q+1, 6q+3 or 6q+5.

Find HCF and LCM of 6, 72 and 120 by prime factorisation method.

The decimal expansion of 441/(2²×5⁷×7²) is terminating or non-terminating?

Prove that √p + √q is irrational where p and q are distinct primes.

Find the greatest number which divides 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.

Prove that √5 is irrational.

Use Euclid's division algorithm to find HCF of 4052 and 12576.

Write whether 2√45 + 3√20 / 2√5 is rational or irrational.

Find LCM and HCF of 336 and 54 by prime factorisation. Verify LCM × HCF = product of numbers.

If p is prime, prove that √p is irrational.