What are the factors of 216 and how do you find them?
The concept of Factors of 216 plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding factors is helpful for solving number system problems, HCF, LCM, and even algebraic questions in competitive exams.
What Are the Factors of 216?
A factor of 216 is any whole number that divides 216 completely without leaving a remainder. Factors of 216 are used in prime factorization, finding HCF and LCM, and checking divisibility in MCQs. For example, factors help in arranging objects, dividing items in equal groups, or checking number patterns and divisibility rules.
List of Factors of 216
Here is a complete list of all the factors of 216, arranged in order:
- 1
- 2
- 3
- 4
- 6
- 8
- 9
- 12
- 18
- 24
- 27
- 36
- 54
- 72
- 108
- 216
Key Formula for Factors of 216
When you have the prime factorization, you can find the total number of factors using:
\( \text{If }216 = 2^3 \times 3^3, \text{ then total factors} = (3+1)\times(3+1) = 16 \)
Prime Factorization of 216
Prime factorization means expressing 216 as a product of only its prime numbers. Let’s break it down step by step:
- 216 ÷ 2 = 108
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
So, 216 = 2 × 2 × 2 × 3 × 3 × 3 = 23 × 33
Factor Pairs of 216
| Pair | Product |
|---|---|
| 1 × 216 | 216 |
| 2 × 108 | 216 |
| 3 × 72 | 216 |
| 4 × 54 | 216 |
| 6 × 36 | 216 |
| 8 × 27 | 216 |
| 9 × 24 | 216 |
| 12 × 18 | 216 |
How to Find the Factors of 216?
You can find the factors of 216 using these simple steps:
1. Start with 1 and the number itself, 216.2. Check every number from 2 upwards: If it divides 216 exactly (no remainder), it is a factor.
3. Continue up to the square root of 216 (around 14.7). After that, pairs begin repeating.
4. Write each factor found in ascending order.
Speed Trick for Factors of 216
Here’s a quick check: If a number’s digits add up to a multiple of 3 or 9, the number is divisible by 3 or 9. For example, 2+1+6=9, so 216 is divisible by both 3 and 9. Use such divisibility rules to quickly check and list factors, saving time in exams!
Solved Example
Q: What is the greatest factor of 216, apart from 216 itself?
1. List all factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216.2. Ignore 216. The next largest is 108.
Final Answer: 108 is the greatest factor of 216 other than 216.
Practice Problems (Try These Yourself!)
- How many even factors does 216 have?
- Find all the factors of 216 between 10 and 40.
- Express 216 as a product of its prime factors.
- Is 54 a factor of 216? Show the step.
Frequent Errors and Misunderstandings
- Missing out factor pairs or skipping factors between 10–40.
- Confusing multiples with factors of 216.
- Thinking 0 is a factor (it’s not!)
Relation to Other Concepts
The idea of factors of 216 connects closely with LCM, HCF, and Prime Factorization. Learning factors lays the foundation for number theory and perfect square or cube problems, which are common in mathematics competitions.
Classroom Tip
A helpful rule: Every whole number, including 216, always has 1 and itself as factors. For composite numbers like 216, start with 2, 3, and so on, to quickly list all factors. Vedantu’s teachers often draw a factor tree diagram on the board to make this topic visual and fun.
We explored Factors of 216 — definition, formula, lists, pairs, prime factorization, errors, and connections. Practice these and use divisibility tricks to build speed. Continue learning and strengthening your fundamentals with Vedantu’s expert maths resources and live classes.
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