Exercise 3.1 (Expanded)

1. Write all the factors of the following numbers:

How to find factors? Think of all pairs of numbers that multiply to give the number. Each number in the pair is a factor.

(a) 24
\(1 \times 24 = 24\) → factors: 1, 24
\(2 \times 12 = 24\) → factors: 2, 12
\(3 \times 8 = 24\) → factors: 3, 8
\(4 \times 6 = 24\) → factors: 4, 6

All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

(b) 15
\(1 \times 15 = 15\) → 1, 15
\(3 \times 5 = 15\) → 3, 5

All factors of 15: 1, 3, 5, 15

All factors of 21: 1, 3, 7, 21

(d) 27
\(1 \times 27 = 27\) → 1, 27
\(3 \times 9 = 27\) → 3, 9

All factors of 27: 1, 3, 9, 27

(e) 12
\(1 \times 12 = 12\) → 1, 12
\(2 \times 6 = 12\) → 2, 6
\(3 \times 4 = 12\) → 3, 4

All factors of 12: 1, 2, 3, 4, 6, 12

(f) 20
\(1 \times 20 = 20\) → 1, 20
\(2 \times 10 = 20\) → 2, 10
\(4 \times 5 = 20\) → 4, 5

All factors of 20: 1, 2, 4, 5, 10, 20

(g) 18
\(1 \times 18 = 18\) → 1, 18
\(2 \times 9 = 18\) → 2, 9
\(3 \times 6 = 18\) → 3, 6

All factors of 18: 1, 2, 3, 6, 9, 18

(h) 23
\(1 \times 23 = 23\) → 1, 23
(No other whole numbers multiply to 23!)

All factors of 23: 1, 23 → 23 is a prime number!

(i) 36
\(1 \times 36 = 36\) → 1, 36
\(2 \times 18 = 36\) → 2, 18
\(3 \times 12 = 36\) → 3, 12
\(4 \times 9 = 36\) → 4, 9
\(6 \times 6 = 36\) → 6 (only once!)

All factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

2. Write first five multiples of:

How to find multiples? Multiply the number by 1, 2, 3, 4, and 5.

(a) 5
\(5 \times 1 = 5\), \(5 \times 2 = 10\), \(5 \times 3 = 15\), \(5 \times 4 = 20\), \(5 \times 5 = 25\)

First five multiples: 5, 10, 15, 20, 25

(b) 8
\(8 \times 1 = 8\), \(8 \times 2 = 16\), \(8 \times 3 = 24\), \(8 \times 4 = 32\), \(8 \times 5 = 40\)

First five multiples: 8, 16, 24, 32, 40

First five multiples: 9, 18, 27, 36, 45

3. Match the items in Column A with the correct items in Column B.

Draw arrows from each item in Column A to its correct match in Column B.

Column A
(i) 35
(ii) 15
(iii) 16
(iv) 20
(v) 25

Column B
(a) Multiple of 8
(b) Multiple of 7
(c) Multiple of 70
(d) Factor of 30
(e) Factor of 50
(f) Factor of 20

Explanation:

(i) 35 = 7 × 5 → multiple of 7 → matches (b)

(ii) 15 divides 30 exactly (30 ÷ 15 = 2) → factor of 30 → matches (d)

(iii) 16 = 8 × 2 → multiple of 8 → matches (a)

(iv) 20 = 4 × 5 → multiple of 20 → matches (f)

(v) 25 divides 50 exactly (50 ÷ 25 = 2) → factor of 50 → matches (e)

Final Matches:
(i) → (b)
(ii) → (d)
(iii) → (a)
(iv) → (f)
(v) → (e)
(Note: (c) Multiple of 70: has no match from Column A)

4. Find all multiples of 9 up to 100.

We multiply 9 by 1, 2, 3, ... until the product is ≤ 100.

\(9 \times 1 = 9\)
\(9 \times 2 = 18\)
\(9 \times 3 = 27\)
\(9 \times 4 = 36\)
\(9 \times 5 = 45\)
\(9 \times 6 = 54\)
\(9 \times 7 = 63\)
\(9 \times 8 = 72\)
\(9 \times 9 = 81\)
\(9 \times 10 = 90\)
\(9 \times 11 = 99\)
\(9 \times 12 = 108\) → too big! (greater than 100)

All multiples of 9 up to 100 are:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99