Question 5 of 18intermediate🔍 AnalyzeShort Answer2 marks

Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6×126 \times 12 and 144 = 8×188 \times 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

Correct Answer

No, one cannot conclude that 72 and 144 have no common factor other than 1.

Reason: The factorisation 72 = 6×126 \times 12 and 144 = 8×188 \times 18 uses composite numbers that can be further factorised.

Prime Factorisation: 72 = 2³ × 3² 144 = 2⁴ × 3²

Common Prime Factors: 2 and 3

HCF: 2³ × 3² = 72

∴ The factorisation into composite numbers hides the true common factors.

Exercise: Figure it Out (Section 3) | Q: 5 | (Page Page 8)
For More Understanding

Explanation

The chapter explains that to find common factors, we need to look at prime factorisations, not just composite factorisations. The numbers 6, 12, 8, and 18 share common prime factors (2 and 3). When factorised completely into primes, 72 = 2³ × 3² and 144 = 2⁴ × 3², showing they share many common factors including 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

Solution Steps

  1. Identify that 6, 12, 8, and 18 are composite numbers that can be further factorised.

  2. Find the prime factorisation of 72 = 2³ × 3².

  3. Find the prime factorisation of 144 = 2⁴ × 3².

  4. Identify common prime factors: 2 and 3.

  5. Calculate HCF = 2³ × 3² = 72.

  6. Conclude that composite factorisation hides the true common factors.

PreviousNext