Step-by-Step Guide to Finding Factor Pairs with Divisibility Examples
The topic of Finding Factor Pairs Using Divisibility Rule is a vital arithmetic skill for students, especially when learning about factors, multiples, and number theory. Mastering this ability helps students excel in school maths, competitive exams like JEE Main, NTSE, and Olympiads, and speeds up mental calculations in daily life and academics.
What Are Factor Pairs and The Divisibility Rule?
A factor pair is a set of two whole numbers that, when multiplied together, give a particular number. For example, (2, 12) is a factor pair of 24, because 2 × 12 = 24. Understanding factor pairs helps students in problem-solving and lays the foundation for advanced mathematical concepts like HCF, LCM, and prime factorization.
The divisibility rule is a shortcut or test used to determine whether one number divides into another without a remainder, thus quickly identifying possible factors. Using divisibility rules makes finding factor pairs much faster, especially for large numbers.
Understanding Factors and Factor Pairs
A factor of a number is any whole number that divides it exactly, leaving no remainder. Factor pairs are simply two such numbers whose product is the target number. For instance, the factor pairs of 18 are (1, 18), (2, 9), and (3, 6).
Unlike multiples, which are obtained by multiplying a number, factors must always divide the original number. It is also worth noting that for every factor pair, the larger number is always matched with its corresponding smaller factor. For visual learners, imagine laying out items in an array—each configuration represents a factor pair.
Divisibility Rules 2 to 11
Here are common divisibility rules that make finding factor pairs much faster:
| Number | Divisibility Rule | Example |
|---|---|---|
| 2 | Last digit is even | 38 (ends in 8) |
| 3 | Sum of digits is divisible by 3 | 123 (1+2+3=6) |
| 4 | Last two digits form a number divisible by 4 | 512 (12 ÷ 4=3) |
| 5 | Ends in 0 or 5 | 85, 40 |
| 6 | Divisible by both 2 and 3 | 132 |
| 8 | Last three digits divisible by 8 | 1,288 (288 ÷ 8=36) |
| 9 | Sum of digits divisible by 9 | 729 (7+2+9=18) |
| 10 | Ends in 0 | 70, 130 |
| 11 | Alternating sum and difference of digits divisible by 11 | 275: (2-7+5)=0 |
How to Find Factor Pairs Using Divisibility Rule?
To find factor pairs using divisibility rules, follow these steps:
- Start with 1 and the given number as your first factor pair.
- Test each number from 2 up to the square root of the target number.
- If a number divides the target number with no remainder (using the divisibility rules), write down both the divisor and the quotient as a factor pair.
- Continue until you reach the square root of the number—after this, factor pairs repeat in reverse.
- List all unique pairs.
Worked Examples
Example 1: Find Factor Pairs of 36 Using Divisibility Rules
- Start: 1 × 36 = 36 (Pair: 1, 36)
- Is 2 a factor? Yes, last digit is even. 36 ÷ 2 = 18 (Pair: 2, 18)
- Is 3 a factor? Sum of digits = 3+6=9, which is divisible by 3. 36 ÷ 3 = 12 (Pair: 3, 12)
- Is 4 a factor? Last two digits are 36; 36 ÷ 4 = 9 (Pair: 4, 9)
- Is 5 a factor? No, does not end in 0 or 5.
- Is 6 a factor? 36 is even and sum is 9 (divisible by 3), so yes. 36 ÷ 6 = 6 (Pair: 6, 6)
Complete list: (1,36), (2,18), (3,12), (4,9), (6,6).
Example 2: Find Factor Pairs of 60
- (1, 60) — 1 always works.
- (2, 30) — 60 ends in 0 (even).
- (3, 20) — 6+0=6, divisible by 3.
- (4, 15) — 60, last two digits 60÷4=15.
- (5, 12) — ends in 0.
- (6, 10) — both 2 and 3 (divisible).
All factor pairs: (1,60), (2,30), (3,20), (4,15), (5,12), (6,10).
Example 3: Find Factor Pairs of 105
- (1, 105)
- (3, 35) — 1+0+5=6 (divisible by 3)
- (5, 21) — ends in 5
- (7, 15) — 105÷7=15
Factor pairs: (1,105), (3,35), (5,21), (7,15).
Practice Problems
- Find all factor pairs of 28 using divisibility rules.
- What are the factor pairs of 72?
- Find the missing pair: 4 × ____ = 64.
- List all factor pairs of 90.
- Which divisibility rules would help you factor 54?
- Is 7 a factor of 56? State the pair.
- Find the factor pairs for 45.
Common Mistakes to Avoid
- Forgetting to check up to the square root only—pairs repeat after that.
- Confusing factors with multiples; factors must divide evenly.
- Missing divisibility rules for numbers like 7 or 11, leading to skipped pairs.
- Leaving out 1 and the number itself as a pair.
- Listing reversed duplicates, e.g., (2,18) and (18,2) for 36.
Real-World Applications
Finding factor pairs is useful beyond exams! It helps in arranging objects, dividing items into equal groups, and designing objects with precise dimensions. Factorization is also fundamental in cryptography, computer science, and in solving equations in engineering. At Vedantu, we show students how these maths skills apply in daily life and higher studies.
In this topic, you learned how to find factor pairs using divisibility rules, understood common shortcuts, saw step-by-step examples, and practiced real problems. Mastering these skills strengthens your arithmetic foundation for both school and competitive exams. For deeper learning, explore related topics like Prime Numbers, Common Factors, and Prime Factorization on Vedantu.
FAQs on How to Find Factor Pairs Using Divisibility Rules
1. How to find factors using divisibility rules?
Divisibility rules offer shortcuts for finding factors. Test numbers up to the square root of your target number. If a number divides evenly (no remainder), it's a factor, and its quotient is its factor pair. For example, with 36, 2 is a factor (36/2=18), and 18 is its pair.
2. What is the divisibility rule of 2, 3, 4, 5, 6, 8, 9, 10, 11?
Each number has a specific divisibility rule: 2 (even last digit), 3 (sum of digits divisible by 3), 4 (last two digits divisible by 4), 5 (ends in 0 or 5), 6 (divisible by both 2 and 3), 8 (last three digits divisible by 8), 9 (sum of digits divisible by 9), 10 (ends in 0), 11 (alternating sum/difference of digits divisible by 11).
3. What is the common factor of 12, 18, and 24 using the divisibility rule for 6?
To find the common factor using the divisibility rule for 6, check if each number (12, 18, 24) is divisible by 6. 12/6=2, 18/6=3, 24/6=4. Since all are divisible by 6, the common factor is 6.
4. How is 275 divisible by 11?
The divisibility rule for 11 involves alternating sums/differences of digits. For 275: 5 - 7 + 2 = 0. Since 0 is divisible by 11, 275 is divisible by 11.
5. What are factor pairs of 60?
Factor pairs of 60 are pairs of numbers that multiply to 60. Using divisibility rules, we find: 1 x 60, 2 x 30, 3 x 20, 4 x 15, 5 x 12, 6 x 10.
6. Finding factor pairs using divisibility rule examples?
Let's find factor pairs of 72. Applying divisibility rules, we find factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The pairs are: 1x72, 2x36, 3x24, 4x18, 6x12, 8x9.
7. What are the divisibility rules for finding factors?
Divisibility rules help quickly identify factors. For example, if a number ends in 0 or 5, it's divisible by 5; if the sum of its digits is divisible by 3, the number is divisible by 3, and so on. These rules streamline factor identification.
8. How do I find common factors using divisibility rules?
To find common factors, apply divisibility rules to each number. Identify the numbers that divide evenly into all numbers. Those are your common factors. For example, the common factors of 12 and 18 are 1, 2, 3, and 6.
9. What are the factors of a number?
Factors of a number are whole numbers that divide the number evenly (without a remainder). For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
10. Why do we stop checking at the square root when finding factor pairs?
When finding factor pairs, you only need to check up to the square root of the number. After that point, the factor pairs simply reverse (e.g., for 36, after checking 6, the remaining pairs are the reverse of the ones before: 6 x 6, then 9 x 4, 12 x 3 and so on).
