Step by Step Method to Find Factor Pairs with Divisibility Rules and 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. What are factor pairs in Maths?
Factor pairs are two numbers that multiply together to give a specific number as their product. In other words, if a × b = n, then (a, b) is a factor pair of n.
- For example, for 12: 1 × 12, 2 × 6, and 3 × 4.
- So the factor pairs of 12 are (1,12), (2,6), and (3,4).
- Factor pairs are always whole numbers when working with positive integers.
2. How do you find factor pairs using divisibility rules?
To find factor pairs using divisibility rules, divide the number by smaller integers and check if the division leaves no remainder. A number n is divisible by another number if the remainder is 0.
- Start from 1 and go up to √n.
- Apply divisibility rules (for 2, 3, 5, 10, etc.).
- If n ÷ a is a whole number, then (a, n ÷ a) is a factor pair.
3. What is the divisibility rule for 2, 3, 5, and 10?
The divisibility rules for 2, 3, 5, and 10 help quickly identify factors without long division.
- Divisible by 2: Last digit is 0, 2, 4, 6, or 8.
- Divisible by 3: Sum of digits is divisible by 3.
- Divisible by 5: Last digit is 0 or 5.
- Divisible by 10: Last digit is 0.
4. How do you find all factor pairs of a number step by step?
To find all factor pairs, divide the number systematically starting from 1 up to its square root. Follow these steps:
- Step 1: Write the number, for example 24.
- Step 2: Divide 24 by 1, 2, 3, 4, etc.
- Step 3: Record pairs where the remainder is 0.
5. Can you give an example of finding factor pairs using divisibility rules?
Yes, for example, to find factor pairs of 30, apply divisibility rules and division.
- 30 ÷ 1 = 30 → (1,30)
- 30 is even → 30 ÷ 2 = 15 → (2,15)
- Sum of digits (3+0=3) → divisible by 3 → 30 ÷ 3 = 10 → (3,10)
- Ends in 0 → divisible by 5 → 30 ÷ 5 = 6 → (5,6)
6. Why do we only check up to the square root when finding factor pairs?
We only check up to the square root because factor pairs repeat after that point in reverse order. If a × b = n and a ≤ √n, then b ≥ √n.
- For 36, √36 = 6.
- Once you reach 6, the pairs start repeating.
- This avoids duplicate factor pairs.
7. What is the difference between factors and factor pairs?
Factors are individual numbers that divide a number exactly, while factor pairs are two factors multiplied together to give the number. For example, for 20:
- Factors: 1, 2, 4, 5, 10, 20
- Factor pairs: (1,20), (2,10), (4,5)
8. How do divisibility rules help in finding factors quickly?
Divisibility rules help identify factors quickly by checking patterns in digits instead of performing full division. For example:
- If a number ends in an even digit, it is divisible by 2.
- If the digit sum is divisible by 3, the number is divisible by 3.
9. Can a number have an odd number of factor pairs?
Yes, a number has an odd number of factors (and a repeated middle pair) if it is a perfect square. For example:
- 16 = 1×16, 2×8, 4×4
- Here, (4,4) is a repeated pair.
10. What are common mistakes when finding factor pairs?
Common mistakes when finding factor pairs include missing factors and not checking divisibility correctly. Students often:
- Stop too early before reaching √n.
- Forget to include 1 and the number itself.
- Make errors in applying divisibility rules.
