How to Find Factors and Prime Status of 31 Easily
The concept of factors of 31 is essential in mathematics and helps students understand divisibility, prime numbers, and number properties, which are useful for exams and problem-solving in class.
Understanding Factors of 31
A factor of 31 is any whole number that divides 31 exactly without leaving a remainder. The factors of a number represent its exact divisors. This concept is widely used in arithmetic, number theory, and finding common factors or multiples with other numbers.
Is 31 a Prime Number?
Yes, 31 is a prime number. A prime number has only two distinct positive factors: 1 and itself. In the case of 31, no number other than 1 and 31 divides it exactly, proving that it’s prime and not composite.
Stepwise Method: How to Find the Factors of 31
To find the factors of 31, follow these steps:
1. Start with 1. Divide 31 by 1. The result is 31, remainder 0, so 1 is a factor.2. Check 2: 31 ÷ 2 = 15.5 (Not an integer, so 2 is not a factor).
3. Check all integers up to 31:
– 31 ÷ 3 = 10.33... (Not a factor)
– 31 ÷ 4 = 7.75 (Not a factor)
– 31 ÷ 5 = 6.2 (Not a factor)
...
4. The final integer is 31 itself. 31 ÷ 31 = 1, remainder 0, so 31 is a factor.
No other whole numbers up to 31 divide 31 exactly, confirming its factors.
All Factors of 31 Table
Here’s a helpful table to understand all the factors of 31:
Factors of 31 Table
| Number | Divides 31? | Result |
|---|---|---|
| 1 | Yes | 31 ÷ 1 = 31 |
| 2 | No | 31 ÷ 2 = 15.5 |
| 3 | No | 31 ÷ 3 ≈ 10.33 |
| ... (4 - 30) | No | Not whole |
| 31 | Yes | 31 ÷ 31 = 1 |
This table shows clearly that only 1 and 31 are factors of 31.
Pair Factors of 31
Factor pairs are pairs of numbers, when multiplied, give the original number. For 31:
1. 1 × 31 = 312. (Also, -1 × -31 = 31 is a negative pair used in advanced maths)
So, the only positive pair factor is (1, 31).
Prime Factorization of 31
Since 31 is a prime number, its prime factorization is simply 31 itself. You can express it as:
There are no other prime factors.
Factors of Nearby Numbers
For comparison, here are factors of nearby numbers:
| Number | All Factors |
|---|---|
| 30 | 1, 2, 3, 5, 6, 10, 15, 30 |
| 31 | 1, 31 |
| 32 | 1, 2, 4, 8, 16, 32 |
| 33 | 1, 3, 11, 33 |
See Factors of 32 and Factors of 30 for more details on nearby numbers.
Worked Example – Are 1 and 31 the Only Factors?
Let’s check if 1 and 31 are the only factors:
1. Divide 31 by each number from 1 to 31 (see table above).2. Only 1 and 31 give integer answers. All others leave a remainder.
3. Therefore, 1 and 31 are the only factors of 31.
Final Answer: Yes, 1 and 31 are the only factors of 31.
Practice Problems
- List all the factors of 31.
- Is 9 a factor of 31? Prove your answer.
- What are the common factors of 31 and 62?
- Is 31 a composite or a prime number?
Common Mistakes to Avoid
- Thinking that 31 has more than two factors.
- Confusing factors with multiples.
- Listing the number 0 as a factor—0 is never a factor.
Where Are Factors of 31 Used?
Factors of 31 are useful in HCF and GCD problems, identifying prime numbers, and divisibility checks in exams. These skills help in Highest Common Factor calculations and competitive maths exams. Knowing how to find factors also builds the foundation for advanced concepts like LCM and prime factorization.
Summary: Key Points on Factors of 31
– The factors of 31 are 1 and 31.
– 31 is a prime number.
– Prime factorization: 31 itself.
– Factor pair: (1, 31).
– Practice identifying factors to prepare for exam questions.
– For more help, visit Factors of a Number or use other Vedantu maths resources.
FAQs on Factors of 31 Explained for Students
1. What are the factors of 31?
The factors of 31 are the numbers that divide 31 exactly without leaving a remainder. Since 31 is a prime number, it has only two factors: 1 and 31 itself.
2. Is 31 a prime or composite number?
The number 31 is a prime number because it has exactly two distinct factors: 1 and 31. It cannot be divided evenly by any other number.
3. What is the highest common factor (HCF) of 31 and 1?
The HCF of 31 and 1 is 1 since 1 is the only number that divides both 31 and 1 exactly without any remainder.
4. How do you find the factors of 31 by division method?
To find the factors of 31 using the division method, divide 31 by consecutive integers starting from 1. If the division leaves a remainder of 0, the divisor is a factor. Since only division by 1 and 31 leaves zero remainder, those are the factors.
5. What are the multiples of 31?
The multiples of 31 are numbers you get by multiplying 31 with whole numbers, such as 31, 62, 93, 124, etc. Multiples are different from factors, as factors divide the number, whereas multiples are products involving the number.
6. Is 9 a factor of 31?
No, 9 is not a factor of 31 because when 31 is divided by 9, it leaves a remainder. Thus, 9 does not divide 31 exactly.
7. Why does 31 only have two factors?
31 has only two factors because it is a prime number. By definition, prime numbers have exactly two distinct factors: 1 and the number itself. No other number divides a prime number exactly.
8. Why can't 31 be divided by a number other than 1 and itself?
31 cannot be divided evenly by any other number except 1 and 31 because it is a prime number. This means no other integer apart from these factors divides 31 without leaving a remainder.
9. Why do students confuse factors with multiples regarding 31?
Students often confuse factors and multiples because both involve division and multiplication. For 31, factors are numbers dividing 31 exactly (1 and 31), while multiples are numbers obtained by multiplying 31 with other integers (31, 62, 93, etc.). Understanding the difference is crucial for exams.
10. How can factors of 31 help in calculating HCF or LCM with other numbers?
Knowing the factors of 31 helps in finding the Highest Common Factor (HCF) and Least Common Multiple (LCM) with other numbers. Since 31 is prime, its only factors are 1 and 31, so the HCF with another number is either 1 or 31 if the other number is a multiple of 31. This simplifies HCF/LCM calculations.
11. Why is divisibility check important for prime numbers like 31?
Divisibility checks are important for prime numbers like 31 to confirm their primality and understand their factor structure. Since prime numbers have only two factors, checking divisibility helps students identify factors accurately, aiding in number theory and exam preparation.
