Question 7 of 37intermediate🔍 AnalyzeMCQ2 marks

Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?

Exercise: Figure it Out (Section 3) | Q: 10 | (Chapter: Page 9)
For More Understanding

Explanation

Find prime factorizations: 45 = 3² × 5 (two distinct primes). 60 = 2² × 3×53 \times 5 (three distinct pr \times but with powers). 91 = 7×137 \times 13 (two distinct primes). 105 = 3×53 \times 5 × 7 (exactly three distinct primes, each appearing once). 330 = 2×32 \times 3 × 5×115 \times 11 (four distinct primes). Only 105 is the product of exactly three distinct prime numbers.

Solution Steps

  1. Step 1: 45 = 3×33 \times 3 × 5 = 3² × 5 (two distinct primes: 3 and 5)

  2. Step 2: 60 = 2×22 \times 2 × 3×53 \times 5 = 2² × 3×53 \times 5 (three distinct pr \times but 2 appears twice)

  3. Step 3: 91 = 7×137 \times 13 (two distinct primes)

  4. Step 4: 105 = 3×53 \times 5 × 7 (exactly three distinct primes, each appearing once) ✓

  5. Step 5: 330 = 2×32 \times 3 × 5×115 \times 11 (four distinct primes)

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