Write 168 as a product of its prime factors. You must show your working (b) Find the highest common factor (HCF) of 168 and 180.
The Correct Answer and Explanation is:
(a) Prime Factorization of 168
To express 168 as a product of its prime factors:
168 ÷ 2 = 84
84 ÷ 2 = 42
42 ÷ 2 = 21
21 ÷ 3 = 7
7 ÷ 7 = 1
Therefore, the prime factorization of 168 = 2 × 2 × 2 × 3 × 7, or in index form:
168 = 2³ × 3 × 7
(b) Find the HCF of 168 and 180
First, find the prime factorization of 180:
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1
So, 180 = 2 × 2 × 3 × 3 × 5, or in index form:
180 = 2² × 3² × 5
Now compare the prime factorizations:
- 168 = 2³ × 3 × 7
- 180 = 2² × 3² × 5
To find the highest common factor (HCF), take the lowest power of common prime factors:
- Common prime factors: 2 and 3
- Minimum power of 2: 2²
- Minimum power of 3: 3¹
So, HCF = 2² × 3 = 4 × 3 = 12
Explanation
Prime factorization involves breaking down a number into its basic building blocks—prime numbers. These are the smallest possible divisors other than one and the number itself. Starting with the smallest prime, division continues step-by-step until reaching one. For the number 168, repeated division by 2 reduces the value until an odd number remains. Further division by 3 and finally 7 completes the process. The result is a multiplication of prime numbers: 2³ × 3 × 7.
For 180, the same method applies. It begins with division by 2, the smallest prime, and continues with 3, followed by 5. Once reduced to 1, the full set of prime factors appears: 2² × 3² × 5. Each of these factors is a building block that, when multiplied, recreates the original number.
To determine the highest common factor (HCF), only the prime numbers that appear in both factorizations are considered. From the lists for 168 and 180, the common primes are 2 and 3. For each common prime, the smallest exponent in the two factorizations is chosen. These minimum powers ensure that the result divides both numbers exactly, but no larger number will.
In this case, 2 appears at most twice (2²), and 3 appears once (3¹) in both numbers. Their product, 2² × 3, equals 12. This value is the greatest number that divides both 168 and 180 without leaving a remainder.
