What Are the Prime Factors and Factor Pairs of 77?
The concept of factors of 77 is a key building block in mathematics, helping students understand divisibility, prime factorization, HCF/LCM, and number properties. Quickly recalling factor lists and their pairs is essential for exam revision and real-life applications like grouping or dividing objects evenly.
What Are Factors of 77?
A factor of 77 is any integer that divides 77 exactly, with no remainder. In other words, if you can multiply two whole numbers to get 77, both numbers are factors of 77. Factors are not the same as multiples; factors are always less than or equal to the number itself. Common related ideas include divisors of 77, factor pairings, and prime factorization.
List of Factors of 77
The complete list of factors of 77 is:
- 1
- 7
- 11
- 77
Each of these numbers divides 77 evenly, leaving no remainder. Negative factors also exist (e.g., -1, -7, -11, -77), but in most maths problems and exams, only positive factors are required.
Prime Factorization of 77
To find the prime factors of 77, break it down into numbers that are only divisible by 1 and themselves:
2. 77 ÷ 7 = 11 (both 7 and 11 are prime numbers).
3. So, 77 = 7 × 11.
So, the prime factorization of 77 is 7 × 11.
| Step | Result |
|---|---|
| Divide by 7 (prime) | 77 ÷ 7 = 11 |
| Divide by 11 (prime) | 11 ÷ 11 = 1 |
Factor Pairs of 77
Factor pairs are pairs of numbers that multiply to 77:
| Pair | Product |
|---|---|
| (1, 77) | 1 × 77 = 77 |
| (7, 11) | 7 × 11 = 77 |
Negative pairs such as (-1, -77) and (-7, -11) are also valid, since multiplying two negatives gives a positive.
Properties and Number Type
- 77 has more than two factors (1, 7, 11, 77), so it is a composite number (not prime).
- It is an odd number and not divisible by 2.
- The sum of all positive factors: 1 + 7 + 11 + 77 = 96.
- The highest common factor (HCF) of 77 and 56 is 7. Learn more: HCF of Two Numbers
- Understanding the difference between factors and multiples often helps speed up MCQ solving.
How to Find Factors of 77: Step-by-Step
2. Test 2 to 6: Not a whole number, so not factors
3. Divide by 7: 77 ÷ 7 = 11 (both 7 and 11 are factors)
4. Test 8 and onwards up to √77 (<9): No more whole-number results
5. List all found: 1, 7, 11, 77
Checking up to the square root ensures results are not repeated.
Speed Trick: Fast Checking for Factors
Exam tip: When asked "Is 77 a prime number?" just list up to √77 (about 8.7). Finding another divisor (like 7 or 11) means it's composite. Use divisibility rules for 7 and 11 for fast checking.
Practice Problems: Factors of 77
Try solving these problems with clear steps:
-
Find all the factors of 77.
1. List divisors of 77 up to 9
2. 77 ÷ 1 = 77
3. 77 ÷ 7 = 11
4. 77 ÷ 11 = 7
5. 77 ÷ 77 = 1
6. Answer: 1, 7, 11, 77 -
What is the prime factorization of 77?
1. Find smallest prime: 7
2. 77 ÷ 7 = 11
3. 11 is prime. So, 77 = 7 × 11 -
List all factor pairs of 77.
1. 1 × 77 = 77
2. 7 × 11 = 77
3. Answer: (1,77), (7,11)
Common Errors & Revision Tips
- Confusing factors and multiples (e.g., listing numbers larger than 77)
- Missing a factor by not checking up to the square root
- Forgetting factor pairs (especially (-1, -77) and (-7, -11) in advanced situations)
- Mixing up divisibility notions; remember, 77 is odd—not divisible by 2
Vedantu teachers recommend building a table or tree diagram for quick revision.
Relation to Other Maths Concepts
Knowing the factors of 77 helps with learning about prime factors, distinguishing composite numbers from primes, and solving for HCF and LCM in word problems.
Quick Check & Summary Table
| Property | Value |
|---|---|
| Number of Positive Factors | 4 |
| Factors of 77 | 1, 7, 11, 77 |
| Prime Factors | 7, 11 |
| Smallest Factor | 1 |
| Largest Factor | 77 |
Wrapping It All Up
We explored factors of 77 from their definition, listing, prime factor breakdown, and factor pairs, all the way to common exam tricks for quick mental recall. Keep using Vedantu’s live classes and study resources for more number property insights and to master speed calculations for exams!
Suggested Lessons & Related Concepts
FAQs on Factors of 77 Explained with Examples
1. What are the factors of 77?
The factors of 77 are the whole numbers that divide 77 without leaving a remainder. These are: 1, 7, 11, and 77. Remember that factors can be positive or negative, so we also have -1, -7, -11, and -77.
2. What is the prime factorization of 77?
The prime factorization of 77 expresses it as a product of prime numbers. Since 7 and 11 are prime numbers and 7 × 11 = 77, the prime factorization of 77 is 7 × 11.
3. Is 77 a prime number or a composite number?
77 is a composite number. A prime number has only two factors (1 and itself), while a composite number has more than two factors. Since 77 has four factors (1, 7, 11, and 77), it's composite.
4. What are the factor pairs of 77?
Factor pairs of 77 are pairs of numbers that multiply to give 77. These are (1, 77) and (7, 11). Considering negative factors, we also have (-1, -77) and (-7, -11).
5. How do I find the factors of a number like 77?
To find the factors of any number, systematically check for divisibility. Start with 1 and work your way up to the square root of the number (approximately 8.77 for 77). If a number divides evenly, both that number and the result are factors. For 77: 1 divides evenly (77/1=77), 7 divides evenly (77/7=11), and 11 divides evenly (77/11=7). We've found all factors.
6. What is the highest common factor (HCF) of 77 and 56?
To find the HCF of 77 and 56, list the factors of each number and identify the largest factor they share. Factors of 77 are 1, 7, 11, 77. Factors of 56 are 1, 2, 4, 7, 8, 14, 28, 56. The highest common factor is 7.
7. What is the sum of the factors of 77?
The sum of the positive factors of 77 (1 + 7 + 11 + 77) is 96.
8. What is the product of the factors of 77?
The product of the positive factors of 77 (1 × 7 × 11 × 77) is 5929.
9. Can 77 be factored further after finding its prime factors?
No. The prime factorization of 77 is 7 × 11. Prime numbers cannot be factored further into smaller whole numbers.
10. How are factors used in solving word problems?
Understanding factors is crucial for solving problems involving equal sharing, grouping, or finding the largest possible size for identical parts. For example, if you have 77 candies to distribute equally among children, knowing the factors helps determine how many children can receive an even number of candies.
11. Explain the difference between factors and multiples.
Factors are numbers that divide a given number without leaving a remainder. Multiples are numbers obtained by multiplying a given number by whole numbers. For example, the factors of 77 are 1, 7, 11, and 77, while multiples of 77 are 77, 154, 231, and so on.
12. Are there any negative factors of 77?
Yes, every positive factor has a corresponding negative factor. The negative factors of 77 are -1, -7, -11, and -77. This is because a negative number multiplied by a negative number gives a positive result.
