Question 2 of 18intermediate🔍 AnalyzeShort Answer2 marks

After observing a few prime factorisations, Anshu claims "The larger a number is, the longer its prime factorisation will be". What do you think of Anshu's claim?

Correct Answer

Anshu's claim is false.

Counterexample: 96 = 2×22 \times 2 × 2×22 \times 2 × 2×32 \times 3 (6 factors in prime factorisation) 121 = 11×1111 \times 11 (only 2 factors)

Observation: Even though 121 > 96, its prime factorisation is shorter.

Reason: The length of prime factorisation depends on the size of prime factors, not the magnitude of the number. Numbers with large prime factors have shorter factorisations than smaller numbers with small prime factors.

Exercise: Figure it Out (Section 1) | Q: 2 | (Page Page 5)
For More Understanding

Explanation

The chapter explains that statements made without proof are called 'conjectures'. Anshu's claim is a conjecture that can be disproved by finding a counterexample. The chapter shows that 96 = 2×22 \times 2 × 2×22 \times 2 × 2×32 \times 3 and 121 = 11×1111 \times 11. Though 121 > 96, the prime factorisation of 121 is shorter because it uses larger prime factors.

Solution Steps

  1. State that Anshu's claim is false.

  2. Provide counterexample: 96 has 6 factors in prime factorisation, 121 has only 2 factors.

  3. Note that 121 > 96 but has shorter factorisation.

  4. Explain that factorisation length depends on prime factor size, not number magnitude.

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