Real Numbers

4ⁿ

Step 1: A number ends in 0 only if it is divisible by both 2 and 5. Step 2: Check 4ⁿ = (2²)ⁿ = 2²ⁿ. Its prime factorisation contains only the prime 2. Step 3: Since 5 is not a factor of 4ⁿ, it cannot be divisible by 10. Step 4: Therefore, 4ⁿ can never end with 0. Step 5: The other options all contain factor 5 in some form, so they can end in 0.

4

Step 1: Find prime factorisation of 96 and 404. Step 2: 96 = 2⁵ × 3. Step 3: 404 = 4 × 101 = 2² × 101 (since 101 is prime). Step 4: Common prime factor is 2. Smallest power of 2 common to both = 2². Step 5: HCF(96, 404) = 2² = 4. Common mistake: Students might not check if 101 is prime and incorrectly factor it.

√7

Step 1: A number is irrational if it cannot be written as p/q where p, q are integers and q ≠ 0. Step 2: √4 = 2 (rational), √9 = 3 (rational), √(25/4) = 5/2 (rational). Step 3: 7 is not a perfect square, and 7 is a prime number. Step 4: By our theorem, √p is irrational when p is prime. Step 5: Since 7 is prime, √7 is irrational. ✓

15

Step 1: Use the formula HCF × LCM = Product of two numbers. Step 2: 5 × 60 = 20 × other number. Step 3: 300 = 20 × other number. Step 4: Other number = 300 ÷ 20 = 15. Step 5: Verify: HCF(20, 15) = 5 ✓ and LCM(20, 15) = 60 ✓.