Real Numbers

√3 is irrational, √3 cannot be expressed as p/q where p, q are integers, √3 is approximately 1.732

√3 is irrational by the same proof technique used for √2. It cannot be expressed as a fraction p/q. Its decimal representation is non-terminating and non-repeating (≈ 1.732050...).

32

Step 1: For prime factorization 2³ × 3² × 5¹ × 7¹. Step 2: Number of factors = (3+1)(2+1)(1+1)(1+1) = 4 × 3 × 2 × 2 = 48. Wait, let me recalculate: (3+1)(2+1)(1+1)(1+1) = 4 × 3 × 2 × 2 = 48. Actually, the correct answer should be 48, but since that's not an option, let me verify: 4 × 3 × 2 × 2 = 48. There seems to be an error in the options. The correct calculation gives 48 factors.

60

Step 1: We need LCM(12, 15, 20). Step 2: 12 = 2² × 3, 15 = 3 × 5, 20 = 2² × 5. Step 3: LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60.

3

Step 1: Number of factors = (n+1)(2+1)(1+1) = (n+1) × 3 × 2 = 6(n+1). Step 2: Given 6(n+1) = 24. Step 3: n+1 = 4, so n = 3.