Find the prime factorisation of each of the following numbers, leaving your answer in index notation. (a) 8624 (b) 6804 (c) 26 163 (d) 196 000
The Correct Answer and Explanation is:
To find the prime factorisation of a number, we break it down into prime numbers (like 2, 3, 5, 7, 11, etc.) that multiply together to give the original number. When expressing the result in index notation, we use exponents to show how many times a prime number repeats in the factorisation.
(a) 8624
Start dividing by the smallest prime number:
- 8624 ÷ 2 = 4312
- 4312 ÷ 2 = 2156
- 2156 ÷ 2 = 1078
- 1078 ÷ 2 = 539
- 539 ÷ 7 = 77
- 77 ÷ 7 = 11
- 11 ÷ 11 = 1
So, the prime factors:
2⁴ × 7² × 11
(b) 6804
Start with 2:
- 6804 ÷ 2 = 3402
- 3402 ÷ 2 = 1701
Now try 3: - 1701 ÷ 3 = 567
- 567 ÷ 3 = 189
- 189 ÷ 3 = 63
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
So, the prime factors:
2² × 3⁵ × 7
(c) 26163
Check for divisibility by 3:
- 2 + 6 + 1 + 6 + 3 = 18 → divisible by 3
- 26163 ÷ 3 = 8721
- 8721 ÷ 3 = 2907
- 2907 ÷ 3 = 969
- 969 ÷ 3 = 323
Now try 17: - 323 ÷ 17 = 19
So, the prime factors:
3⁴ × 17 × 19
(d) 196000
Start with 10:
196000 = 196 × 1000
Break each part:
- 196 = 2² × 7²
- 1000 = 10³ = (2 × 5)³ = 2³ × 5³
Now multiply all together:
2² × 7² × 2³ × 5³
Combine like bases:
2⁵ × 5³ × 7²
Explanation
Prime factorisation is the process of writing a number as a product of prime numbers. Prime numbers are those greater than 1 with only two factors — 1 and themselves. Every composite number has a unique prime factorisation, which makes this process valuable in arithmetic, algebra, and number theory.
The factorisation is done by dividing the number repeatedly by the smallest prime number possible (starting with 2, then 3, 5, etc.) until we reach 1. Whenever a prime number divides evenly into the number, we record it and divide the result again. If it does not divide evenly, we move to the next higher prime number.
For large numbers, breaking them into smaller known values (like 1000 = 10³) can simplify the work. Once we finish breaking down the number into all its prime components, we collect like bases and express the count of each as an exponent. This is called index notation. For example, instead of writing 2 × 2 × 2 × 2 = 16, we write 2⁴.
Using index notation helps simplify expressions, especially when working with large or repeated prime factors. It also makes it easier to compare factorizations or compute greatest common divisors and least common multiples.
Understanding prime factorisation is also essential in cryptography, data encoding, and solving mathematical problems involving divisibility, fractions, and ratios.
