Exercise 3.4

1. Find the common factors of:
(a) 20 and 28
Factors of 20: 1, 2, 4, 5, 10, 20
Factors of 28: 1, 2, 4, 7, 14, 28
Common factors: 1, 2, 4
(b) 15 and 25
Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
Common factors: 1, 5
(c) 35 and 50
Factors of 35: 1, 5, 7, 35
Factors of 50: 1, 2, 5, 10, 25, 50
Common factors: 1, 5
(d) 56 and 120
Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120
Common factors: 1, 2, 4, 8
2. Find the common factors of:
(a) 4, 8 and 12
Factors of 4: 1, 2, 4
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Common factors: 1, 2, 4
(b) 5, 15 and 25
Factors of 5: 1, 5
Factors of 15: 1, 3, 5, 15
Factors of 25: 1, 5, 25
Common factors: 1, 5
3. Find first three common multiples of:
(a) 6 and 8
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 72, 78, 96, ...
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, ...
Common multiples: 24, 48, 72, 96, ...
First three: 24, 48, 72
(b) 12 and 18
Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...
Multiples of 18: 18, 36, 54, 72, 90, 108, 126, ...
Common multiples: 36, 72, 108, ...
First three: 36, 72, 108
4. Write all the numbers less than 100 which are common multiples of 3 and 4.

A common multiple of 3 and 4 is a multiple of their LCM.
LCM of 3 and 4 = \(3 \times 4 = 12\) (since they are co-prime).

Multiples of 12 less than 100:

\(12 \times 1 = 12\)
\(12 \times 2 = 24\)
\(12 \times 3 = 36\)
\(12 \times 4 = 48\)
\(12 \times 5 = 60\)
\(12 \times 6 = 72\)
\(12 \times 7 = 84\)
\(12 \times 8 = 96\)
\(12 \times 9 = 108\) → too big!
Common multiples: 12, 24, 36, 48, 60, 72, 84, 96
5. Which of the following numbers are co-prime?

Co-prime numbers have only 1 as a common factor.

(a) 18 and 35
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 35: 1, 5, 7, 35
Common factor: 1 → Co-prime
(b) 15 and 37
37 is prime. Factors of 15: 1, 3, 5, 15
Common factor: 1 → Co-prime
(c) 30 and 415
30 = 2×3×5, 415 ends with 5 → divisible by 5 → 415 = 5×83
Common factor: 5 → Not co-prime
(d) 17 and 68
17 is prime. 68 ÷ 17 = 4 → so 17 is a factor of 68
Common factors: 1, 17 → Not co-prime
(e) 216 and 215
These are consecutive numbers → always co-prime!
Common factor: 1 → Co-prime
(f) 81 and 16
81 = \(3^4\), 16 = \(2^4\) → no common prime factors
Common factor: 1 → Co-prime
6. A number is divisible by both 5 and 12. By which other number will that number be always divisible?

If a number is divisible by both 5 and 12, it must be divisible by their LCM.

5 and 12 are co-prime → LCM = \(5 \times 12 = 60\).

So the number is divisible by 60, and therefore by all factors of 60.

But the question asks for **one other number** — the most direct answer is the LCM itself.

The number will always be divisible by 60.
7. A number is divisible by 12. By what other numbers will that number be divisible?

If a number is divisible by 12, it is also divisible by all factors of 12.

Factors of 12: 1, 2, 3, 4, 6, 12

So, the number is also divisible by: 1, 2, 3, 4, 6

Remember: Every number is divisible by its factors!
Other numbers: 1, 2, 3, 4, 6