Factors of 51: Definition, List, Prime Factorization with Solved Examples

The concept of factors of 51 plays a key role in mathematics and is widely applicable to solving divisibility questions, prime factorization, and many real-life problems involving grouping, sharing, and simplification. Understanding the factors of 51 can help students save time during math exams and lay a strong foundation for topics like LCM, HCF, and simplifying fractions.

What Are the Factors of 51?

A factor of 51 is any whole number that divides 51 exactly without leaving a remainder. In other words, if you multiply two whole numbers and get 51 as the answer, those numbers are called the factors of 51. These factors are important for tasks like splitting items evenly, simplifying mathematical expressions, and understanding number properties. Factors of 51 are often used in topics such as multiples, prime factorization, and composite numbers.

All Factors of 51 Listed

The positive factors of 51 are:

• 1
• 3
• 17
• 51

Each of these numbers divides 51 exactly. For example, 3 × 17 = 51 and 1 × 51 = 51.

Step-by-Step: How to Find the Factors of 51

1. Start by dividing 51 by 1. The remainder is 0. So, 1 is a factor.
2. Next, try 2. 51 ÷ 2 = 25.5, not a whole number. So, 2 is not a factor.
3. Try 3. 51 ÷ 3 = 17. 17 is a whole number, so 3 and 17 are both factors.
4. Next, check 4, 5, 6, etc. None of these give a whole number until you reach 17 and 51.
5. Since 17 × 3 = 51 and 1 × 51 = 51, we have found all positive factors.

So, the factors of 51 are 1, 3, 17, and 51.

Key Formula for Factors

For any number n: Find all whole numbers, starting from 1 up to n, that divide n exactly (no remainder).
For 51: Test numbers from 1 to 51 to see which can divide exactly.

Prime Factorization of 51

The prime factorization of 51 means expressing 51 as a product of its prime factors.

1. Start by dividing 51 by the smallest prime, which is 2. 51 ÷ 2 = 25.5 (not exact).
2. Try the next prime, 3. 51 ÷ 3 = 17
3. 3 is prime, and 17 is also prime.
4. Therefore, prime factors of 51 = 3 × 17.

So, the only prime factors of 51 are 3 and 17.

Factor Pairs of 51

Factor pairs are two whole numbers that multiply together to give 51. Both positive and negative pairs exist.

Factor 1	Factor 2
1	51
3	17
17	3
51	1

Negative pairs are also valid, such as (-1, -51) and (-3, -17).

Is 51 a Prime or Composite Number?

51 is a composite number, which means it has more than two factors (1, itself, and others). Since 3 and 17 are also factors of 51, it is not a prime number.

Using Factors of 51 in LCM and HCF

Knowing the factors of 51 helps you quickly find the LCM (least common multiple) and HCF (highest common factor) when working with other numbers for math problems and word problems. For example, the factors of 51 can be used together with the factors of 34 or 52 to find common multiples or greatest common divisors. 

Connections: Multiples and Related Numbers

Multiples of 51 are numbers you get by multiplying 51 by any whole number. The first few multiples are 51, 102, 153, 204, and 255. 

For more about primes versus composites, visit Prime Numbers and compare with the factors of 52 or factors of 34 to spot patterns in consecutive numbers.

Solved Example: Are 3 and 17 Both Factors of 51?

Let’s verify if both 3 and 17 are factors of 51:

1. 51 ÷ 3 = 17, so 3 is a factor.

2. 51 ÷ 17 = 3, so 17 is also a factor.

3. Both multiply to give back 51: 3 × 17 = 51.

Final Answer: Yes, both 3 and 17 are factors of 51.

Trick to Find Factors Quickly

To check if a number like 51 is divisible by 3, add the digits: 5 + 1 = 6, which is divisible by 3. So, 3 is a factor! Try similar tricks for other numbers to speed up exams. Vedantu offers more such tips in live classes!

Try These Yourself

• Find all factors of 52.
• Is 17 a prime factor of 51?
• Give all common factors of 34 and 51.
• List the first five multiples of 51.
• Is 51 a prime number? Why or why not?

Common Mistakes Students Make

• Forgetting to check every number up to the square root of 51 for factors.
• Missing negative factor pairs like -3 and -17.
• Confusing prime factors with all factors.
• Assuming factors and multiples are the same concept.

Classroom Tip

A quick memory trick from Vedantu's teachers: "If a number's digits add up to a number divisible by 3, then 3 is a factor." This saves time when checking larger numbers during rapid calculations in class or for exams.

Wrapping It All Up

We explored factors of 51—covering definitions, calculation steps, pairs, prime factorization, and tips for remembering them. Continue practicing similar problems with Vedantu for increased speed, accuracy, and confidence, especially in topics like HCF, LCM, and prime factorization. For even more practice, check out Prime Factorization and Factors of 100.