Question 22 of 36intermediate🔧 ApplyNumerical5 marks

Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.

Correct Answer

Explanation: Prime factorisation involves breaking down each number into its prime factors.

(a) 64: 64 = 2×322 \times 32 = 2×22 \times 2 × 16 = 2×22 \times 2 × 2×82 \times 8 = 2×22 \times 2 × 2×22 \times 2 × 4 = 2×22 \times 2 × 2×22 \times 2 × 2×22 \times 2 ∴ 64 = 2⁶

(b) 104: 104 = 2×522 \times 52 = 2×22 \times 2 × 26 = 2×22 \times 2 × 2×132 \times 13 ∴ 104 = 2³ × 13

(c) 105: 105 = 3×353 \times 35 = 3×53 \times 5 × 7 ∴ 105 = 3×53 \times 5 × 7

(d) 243: 243 = 3×813 \times 81 = 3×33 \times 3 × 27 = 3×33 \times 3 × 3×93 \times 9 = 3×33 \times 3 × 3×33 \times 3 × 3 ∴ 243 = 3⁵

(e) 320: 320 = 2×1602 \times 160 = 2×22 \times 2 × 80 = 2×22 \times 2 × 2×402 \times 40 = 2×22 \times 2 × 2×22 \times 2 × 20 = 2×22 \times 2 × 2×22 \times 2 × 2×102 \times 10 = 2×22 \times 2 × 2×22 \times 2 × 2×22 \times 2 × 5 ∴ 320 = 2⁶ × 5

(f) 141: 141 = 3×473 \times 47 ∴ 141 = 3×473 \times 47

(g) 1728: 1728 = 2×8642 \times 864 = 2×22 \times 2 × 432 = ... = 2⁶ × 3³ ∴ 1728 = 2⁶ × 3³

(h) 729: 729 = 3×2433 \times 243 = 3×33 \times 3 × 81 = ... = 3⁶ ∴ 729 = 3⁶

(i) 1024: 1024 = 2¹⁰ ∴ 1024 = 2¹⁰

(j) 1331: 1331 = 11×12111 \times 121 = 11×1111 \times 11 × 11 ∴ 1331 = 11³

(k) 1000: 1000 = 2×5002 \times 500 = 2×22 \times 2 × 250 = 2×22 \times 2 × 2×1252 \times 125 = 2³ × 5³ ∴ 1000 = 2³ × 5³

Exercise: Figure it Out (Section 4) | Q: 1 | (Page Page 14)
For More Understanding

Explanation

Prime factorisation involves breaking down each number into its prime factors. For example, 64 = 2×322 \times 32 = 2×22 \times 2 × 16 = 2×22 \times 2 × 2×82 \times 8 = 2×22 \times 2 × 2×22 \times 2 × 4 = 2×22 \times 2 × 2×22 \times 2 × 2×22 \times 2 = 2⁶. Similarly, other numbers are factorised.

Solution Steps

  1. Prime factorisation involves breaking down each number into its prime factors.

  2. For 64: Factorise as 2×322 \times 32 = 2×22 \times 2 × 16 = ... = 2×22 \times 2 × 2×22 \times 2 × 2×22 \times 2 = 2⁶

  3. For 104: Factorise as 2×522 \times 52 = 2×22 \times 2 × 26 = 2×22 \times 2 × 2×132 \times 13 = 2³ × 13

  4. For 105: Factorise as 3×353 \times 35 = 3×53 \times 5 × 7

  5. For 243: Factorise as 3×813 \times 81 = 3×33 \times 3 × 27 = 3×33 \times 3 × 3×93 \times 9 = 3×33 \times 3 × 3×33 \times 3 × 3 = 3⁵

  6. For 320: Factorise as 2×1602 \times 160 = ... = 2×22 \times 2 × 2×22 \times 2 × 2×22 \times 2 × 5 = 2⁶ × 5

  7. For 141: Factorise as 3×473 \times 47

  8. For 1728: Factorise as 2×8642 \times 864 = ... = 2⁶ × 3³

  9. For 729: Factorise as 3×2433 \times 243 = ... = 3⁶

  10. For 1024: Factorise as 2¹⁰

  11. For 1331: Factorise as 11×12111 \times 121 = 11×1111 \times 11 × 11 = 11³

  12. For 1000: Factorise as 2×5002 \times 500 = ... = 2³ × 5³

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