Q1.
List all the factors of the following numbers: (a) 90 (b) 105 (c) 132 (d) 360 (this number has 24 factors) (e) 840 (this number has 32 factors)
Answer
Factorise each number into primes, then build every subpart.
(a) 90 = 2 × 3 × 3 × 5
(b) 105 = 3 × 5 × 7
(c) 132 = 2 × 2 × 3 × 11
(d) 360 = 2 × 2 × 2 × 3 × 3 × 5
(e) 840 = 2 × 2 × 2 × 3 × 5 × 7
(b) 105 = 3 × 5 × 7
(c) 132 = 2 × 2 × 3 × 11
(d) 360 = 2 × 2 × 2 × 3 × 3 × 5
(e) 840 = 2 × 2 × 2 × 3 × 5 × 7
| Number | All factors | How many | |
|---|---|---|---|
| (a) | 90 | 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90 | 12 |
| (b) | 105 | 1, 3, 5, 7, 15, 21, 35, 105 | 8 |
| (c) | 132 | 1, 2, 3, 4, 6, 11, 12, 22, 33, 44, 66, 132 | 12 |
| (d) | 360 | 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360 | 24 |
| (e) | 840 | 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840 | 32 |
Tip — count before you list. Write the factorisation with powers and add 1 to each power, then multiply.
90 = 2¹ × 3² × 5¹ → 2 × 3 × 2 = 12 factors
105 = 3¹ × 5¹ × 7¹ → 2 × 2 × 2 = 8 factors
132 = 2² × 3¹ × 11¹ → 3 × 2 × 2 = 12 factors
360 = 2³ × 3² × 5¹ → 4 × 3 × 2 = 24 factors
840 = 2³ × 3¹ × 5¹ × 7¹ → 4 × 2 × 2 × 2 = 32 factors
The counts for (d) and (e) match what the book says, so nothing has been missed.
90 = 2¹ × 3² × 5¹ → 2 × 3 × 2 = 12 factors
105 = 3¹ × 5¹ × 7¹ → 2 × 2 × 2 = 8 factors
132 = 2² × 3¹ × 11¹ → 3 × 2 × 2 = 12 factors
360 = 2³ × 3² × 5¹ → 4 × 3 × 2 = 24 factors
840 = 2³ × 3¹ × 5¹ × 7¹ → 4 × 2 × 2 × 2 = 32 factors
The counts for (d) and (e) match what the book says, so nothing has been missed.
Why it happens: to build a factor of 360 = 2³ × 3² × 5 you decide how many 2s to keep (0, 1, 2 or 3 — four choices), how many 3s (0, 1 or 2 — three choices) and how many 5s (0 or 1 — two choices). Each different set of choices gives a different factor, so there are 4 × 3 × 2 = 24 of them.