Class 10 Real Numbers - Important Questions with Solutions
1. Use Euclids Division Algorithm to find the HCF of 252 and 198.
252 198 = 1, remainder = 54
198 54 = 3, remainder = 36
54 36 = 1, remainder = 18
36 18 = 2, remainder = 0
HCF = 18
2. Show that 2 is an irrational number.
Assume 2 = a/b, where a, b are coprime.
Then, a^2 = 2b^2 a is even a = 2k b^2 = 2k^2 b is even.
Contradiction as both a and b are even.
So, 2 is irrational.
3. Find the HCF and LCM of 96 and 404 using prime factorization method.
96 = 2^5 x 3, 404 = 2^2 x 101
HCF = 2^2 = 4
LCM = 2^5 x 3 x 101 = 9696
4. Show that 5 is a factor of (3^5 + 2^5 - 1).
3^5 = 243, 2^5 = 32 243 + 32 - 1 = 274
274 5 = 54.8 Not divisible by 5
5. Prove that there are infinitely many prime numbers.
Assume primes are p1, p2, ..., pn
Let P = p1 x p2 x ... x pn + 1
P is not divisible by any of them Contradiction
So, infinite primes exist.
6. Express 23.765 as a rational number.
23.765 = =
1. Use Euclids Division Algorithm to find the HCF of 252 and 198.
252 198 = 1, remainder = 54
198 54 = 3, remainder = 36
54 36 = 1, remainder = 18
36 18 = 2, remainder = 0
HCF = 18
2. Show that 2 is an irrational number.
Assume 2 = a/b, where a, b are coprime.
Then, a^2 = 2b^2 a is even a = 2k b^2 = 2k^2 b is even.
Contradiction as both a and b are even.
So, 2 is irrational.
3. Find the HCF and LCM of 96 and 404 using prime factorization method.
96 = 2^5 x 3, 404 = 2^2 x 101
HCF = 2^2 = 4
LCM = 2^5 x 3 x 101 = 9696
4. Show that 5 is a factor of (3^5 + 2^5 - 1).
3^5 = 243, 2^5 = 32 243 + 32 - 1 = 274
274 5 = 54.8 Not divisible by 5
5. Prove that there are infinitely many prime numbers.
Assume primes are p1, p2, ..., pn
Let P = p1 x p2 x ... x pn + 1
P is not divisible by any of them Contradiction
So, infinite primes exist.
6. Express 23.765 as a rational number.
23.765 = =
