Class V Essentials: Learn Divisibility, Factors, HCF and LCM the Easy Way
Published by Ganit Charcha
| Category - Middle School Math Topics | 2025-07-12 12:02:54
Numbers aren’t just symbols—they’re puzzles waiting to be solved! In Class V, math takes a fun leap as we explore how numbers interact through divisibility, multiples, factors, HCF, and LCM. These core concepts unlock smarter ways to handle big problems, recognize patterns, and think like a true number wizard. Whether you're solving sums or spotting shortcuts, this guide will sharpen your skills and boost your math confidence.
1. Divisibility
A number A is divisible by another number B if, when you divide A by B, there is no remainder.
Notation: We write “B ∣ A” to mean “B divides A.”
Example:
○ 15 ÷ 5 = 3 exactly, so 5 ∣ 15.
○ 14 ÷ 4 = 3 remainder 2, so 4 ∤ 14 (4 does not divide 14).
Quick divisibility checks:
Divisible by 2? Last digit even (0, 2, 4, 6, 8).
Divisible by 5? Last digit is 0 or 5.
Extra for practice:
2: Last digit is even (0, 2, 4, 6, 8).
3: Sum of all digits is a multiple of 3.
4: Last two digits form a number that’s a multiple of 4 (e.g., 312 → 12 is divisible by 4).
5: Last digit is 0 or 5.
6: The number is divisible by both 2 and 3.
7: Take the last digit, double it, and subtract that from the rest of the number. If the result is divisible by 7 (including 0), then the original number is too. Repeat as needed.
Example: 203
■ Last digit = 3 → 3 × 2 = 6
■ Remaining part = 20 → 20 − 6 = 14
■ 14 ÷ 7 = 2 exactly → so 203 is divisible by 7.
8: Last three digits form a number that’s a multiple of 8 (e.g., 5216 → 216 ÷ 8 = 27).
9: Sum of all digits is a multiple of 9.
10: Last digit is 0.
11: Take the alternating sum of the digits (add, subtract, add, subtract…); if the result is a multiple of 11 (including 0), then the number is divisible by 11.
Example: For 3: 574: (3 – 5 + 7 – 4) = 1 → not divisible by 11.
Example: For 2: 310: (2 – 3 + 1 – 0) = 0 → divisible by 11.
2. Multiples
A multiple of a number is what you get when you multiply that number by any whole number.
Example:
a. Multiples of 4: 4 × 1 = 4, 4 × 2 = 8, 4 × 3 = 12, … so 4, 8, 12, 16, … are all multiples of 4.
b. Multiples of 7: 7, 14, 21, 28, …
Practice:List the first five multiples of 6.
Answer: 6, 12, 18, 24, 30.
3. Highest Common Factor (HCF) - Also called the Greatest Common Divisor (GCD) The HCF of two (or more) numbers is the largest number that divides each of them exactly.
How to find it:
List factors of each number.
Circle the common ones.
Pick the largest.
Example: Find HCF of 18 and 24.
○ Factors of 18: 1, 2, 3, 6, 9, 18
○ Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
○ Common factors: 1, 2, 3, 6 ⇒ HCF = 6
Euclid’s Algorithm (using remainders) - Repeatedly subtract or use remainders:
Divide larger by smaller: 24 ÷ 18 = 1 rem 6
Now find HCF(18, 6): 18 ÷ 6 = 3 rem 0 ⇒ stops here.
HCF = 6.
Prime Factorisation Method:
Write each number as a product of primes.
Identify the primes common to both, taking the lowest power of each.
Multiply those together.
Example: Find HCF of 18 and 24.
○ 18 = 2¹ × 3²
○ 24 = 2³ × 3¹
○ Common primes: 2 (use the lower exponent 1), 3 (use the lower exponent 1)
○ HCF = 2¹ × 3¹ = 2 × 3 = 6
Common Division Method
Write the two numbers side by side.
Divide them both by a common prime (or any divisor) until no more common divisors > 1 exist.
Multiply all the divisors used—this product is the HCF.
Example: HCF of 48 and 60.
2 │ 48 60 ← both divisible by 2
2 │ 24 30 ← both again by 2
3 │ 12 15 ← both divisible by 3
4 5
HCF = 2 × 2 × 3 = 12
4. Lowest Common Multiple (LCM)
The LCM of two (or more) numbers is the smallest number (greater than 0) that is a multiple of each of them.
How to find it:
List multiples of each number.
Find the first common one.
Example: LCM of 4 and 6.
○ Multiples of 4: 4, 8, 12, 16, …
○ Multiples of 6: 6, 12, 18, 24, …
○ First common multiple: 12
Prime-factor method:
Prime-factor each number.
For each prime, take the highest power seen.
Multiply them together.
Example: LCM of 18 and 24.
○ 18 = 2¹ × 3²
○ 24 = 2³ × 3¹
○ For prime 2: highest power is 2³
○ For prime 3: highest power is 3²
○ LCM = 2³ × 3² = 8 × 9 = 72
Common Division Method
Write the two numbers side by side.
Divide them both by any common prime (or divisor), writing down the quotient each time; if one number isn’t divisible, carry it down unchanged.
Continue until both become 1.
Multiply all the divisors used—this product is the LCM.
Example: LCM of 8 and 12.
2 │ 8 12 ← both by 2
2 │ 4 6 ← both by 2 again
2 │ 2 3 ← both by 2? no, only 2→1; 3 stays at 3
3 │ 1 3 ← divide both by remaining common 3? only 3→1
1 1
LCM = 2 × 2 x 2 × 3 = 24
5. Divisor Counting (Number of Divisors)
How many positive divisors (factors) does a number have?
Approach:
List all factors and count them.
○ Example: Factors of 12 are 1, 2, 3, 4, 6, 12 ⇒ 6 divisors.
Efficient method (using prime factorisation):
Write the number as a product of primes:
N = p₁ᵃ × p₂ᵇ × p₃ᶜ × …
The total number of positive divisors is:
(a + 1) × (b + 1) × (c + 1) …
Example:How many divisors does 360 have?
Prime factors: 360 = 2³ × 3² × 5¹
Exponents are 3, 2, 1 ⇒ add 1 to each: 4, 3, 2
Multiply: 4 × 3 × 2 = 24 So, 360 has 24 positive divisors.
6. Relationship between HCF and LCM The HCF and LCM of two numbers are closely related through a simple mathematical identity: HCF(a,b) × LCM(a,b) = a × b This means that the product of the HCF and LCM of two numbers is equal to the product of the numbers themselves. Example: Let’s take two numbers: 12 and 18 HCF of 12 and 18 = 6 LCM of 12 and 18 = 36 Therefore, HCF x LCM = 6 x 36 = 216 = 12 x 18 = Product of the Two Numbers
Practice Problems
Divisibility: Is 273 divisible by 3?
Divisibility: Is 462 divisible by 7?
Multiples: Write the first four multiples of 9.
Multiples: List three common multiples of 5 and 8.
HCF: Find the HCF of 28 and 42 (use prime factors).
HCF: Use the common division method to find the HCF of 36 and 54.
LCM: Find the LCM of 9 and 12.
LCM: Use prime‐factor method to find the LCM of 14 and 20.
Counting Divisors: How many positive divisors does 84 have?
Counting Divisors: How many positive divisors does 210 have?
Two numbers have an HCF of 8 and an LCM of 96. If one of the numbers is 24, find the other number.
The product of two numbers is 540. If their HCF is 6, find their LCM.
Answers 1. 2 + 7 + 3 = 12; 12 is divisible by 3 ⇒ Yes 2. 462 → last digit 2, so test 7: 46 − (2×2)=46−4=42; 42 ÷ 7 = 6 ⇒ Yes. 3. 9, 18, 27, 36. 4. Multiples of 5: 5, 10, 15, 20, 25, 30, …; of 8: 8, 16, 24, 32, … ⇒ common: 40, 80, 120 (first three). 5. 28 = 2²×7¹; 42 = 2¹×3¹×7¹ → common 2¹,7¹ → HCF = 2×7 = 14. 6. HCF of 36 and 54
2 │ 36 54
3 │ 18 27
3 │ 6 9
2 3
HCF = 2×3×3 = 18
7. Multiples of 9: 9,18,27,36,…; of 12: 12,24,36,… ⇒ LCM = 36. 8. 14 = 2¹×7¹; 20 = 2²×5¹ → take 2²,5¹,7¹ → LCM = 4×5×7 = 140. 9. 84 = 2²×3¹×7¹ ⇒ (2+1)(1+1)(1+1)=3×2×2 = 12. 10. 210 = 2¹×3¹×5¹×7¹ ⇒ (1+1)(1+1)(1+1)(1+1) = 2×2×2×2 = 16. 11. Product of two numbers = HCF x LCM = 8 x 96 = 768 Therefore, the other number = 768 ÷ 24 = 32. 12. HCF x LCM = Product of two numbers = 540. Given, HCF = 6. Therefore, LCM = 540 ÷ 6 = 90.
Challenges: Learn, Practice, Repeat, Master!
Here are 25 varied practice problems spanning divisibility, multiples, HCF, LCM, and divisor counting. Try to use the methods you’ve learned—divisibility tests, listing, prime factorisation, Euclid’s algorithm, and the common-division method.
Divisibility: Is 1,234 divisible by 2?
Divisibility: Is 1,234 divisible by 3?
Divisibility: Is 1,234 divisible by 4?
Divisibility: Is 1,234 divisible by 5?
Divisibility: Is 1,234 divisible by 6?
Divisibility: Is 1,234 divisible by 8?
Divisibility: Is 1,234 divisible by 9?
Divisibility: Is 1,234 divisible by 10?
Divisibility: Is 1,234 divisible by 11?
Divisibility: Is 1,234 divisible by 7?
Multiples: Write the first five multiples of 13.
Multiples: List three common multiples of 6 and 15.
Multiples: What’s the 10th multiple of 4?
HCF (listing factors): Find the HCF of 45 and 60.
HCF (Euclid’s algorithm): Find the HCF of 107 and 89.
HCF (prime-factor method): Find the HCF of 84 and 126.
HCF (common-division method): Find the HCF of 72 and 108.
LCM (listing multiples): Find the LCM of 8 and 14.
LCM (prime-factor method): Find the LCM of 18 and 30.
LCM (common-division method): Find the LCM of 9 and 12.
Combined HCF & LCM: Given two numbers whose HCF is 6 and LCM is 180, and one number is 54, find the other.
Divisor Counting (small): How many positive divisors does 36 have?
Divisor Counting (medium): How many positive divisors does 720 have?
Divisor Counting (challenge): How many positive divisors does 4,410 have?
Integrated Challenge: Pick any two different two-digit numbers.
Test each for divisibility by 2–11.
Find their HCF by prime-factor and common-division methods.
Find their LCM by prime-factor and common-division methods.
Count how many divisors each has.
Answers to the Challenge 1. Divisible by 2? 1,234 → last digit 4 (even) ⇒ Yes 2 Divisible by 3? 1+2+3+4 = 10, not a multiple of 3 ⇒ No 3. Divisible by 4? Last two digits 34; 34 ÷ 4 = 8 r2 ⇒ No 4. Divisible by 5? Last digit ≠ 0 or 5 ⇒ No 5. Divisible by 6? Must be by 2 and 3; it is even (✔) but not by 3 ⇒ No 6. Divisible by 8? Last three digits 234; 234 ÷ 8 = 29 r2 ⇒ No 7. Divisible by 9? Sum of digits = 10, not a multiple of 9 ⇒ No 8. Divisible by 10? Last digit ≠ 0 ⇒ No 9. Divisible by 11? Alternating sum = 1–2+3–4 = –2, not a multiple of 11 ⇒ No 10. Divisible by 7? 123 – (2×4)=123–8=115; 11 – (2×5)=11–10=1, not a multiple of 7 ⇒ No 11. First five multiples of 13: 13, 26, 39, 52, 65
12. Three common multiples of 6 & 15: ○ Multiples of 6: 6,12,18,24,30,... ○ Multiples of 15: 15,30,45,... ⇒ 30, 60, 90 13. 10th multiple of 4: 4 × 10 = 40
14. HCF of 45 & 60 (listing factors) and
45 = 1,3,5,9,15,45
60 = 1,2,3,4,5,6,10,12,15,20,30,60
common = 1,3,5,15 ⇒ 15
15. HCF of 107 and 89 (Euclid's Algorithm)
107 ÷ 89 = 1 r18
89 ÷ 18 = 4 r17
18 ÷ 17 = 1 r1
17 ÷ 1 = 17 r0 ⇒ HCF = 1
16. HCF of 84 and 126 (Prime Factorization Method)
84 = 2² × 3¹ × 7¹
126 = 2¹ × 3² × 7¹
common lowest exponents = 2¹,3¹,7¹ ⇒ 2×3×7 = 42
17. HCF of 72 & 108 (common‐division)
2 │ 72 108
2 │ 36 54
3 │ 18 27
3 │ 6 9
2 3
HCF = 2×2×3×3 = 36
18. LCM of 8 & 14 (listing multiples)
8 → 8,16,24,32,40,48,56,…
14 → 14,28,42,56,… ⇒ 56
19. LCM of 18 and 30 (Prime Factorization Method)
18 = 2¹ × 3²
30 = 2¹ × 3¹ × 5¹
Take highest exponents: 2¹,3²,5¹ ⇒ 2×9×5 = 90
20. LCM of 9 & 12 (common‐division)
3 │ 9 12 ← divide both by 3
3 │ 3 4 ← divide 3→1; 4 stays 4
4 │ 1 4 ← divide both by 4? 1 stays, 4→1
1 1
LCM = 3×3×4 = 36
21. Given HCF = 6, LCM = 180, one number 54 ⇒ other = (6×180)÷54 = 1,080÷54 = 20 22. Divisors of 36 = 2²×3² ⇒ (2+1)(2+1) = 9 23. Divisors of 720 = 2⁴×3²×5¹ ⇒ (4+1)(2+1)(1+1) = 5×3×2 = 30 24. Divisors of 4,410 = 2¹×3²×5¹×7² ⇒ (1+1)(2+1)(1+1)(2+1) = 2×3×2×3 = 36
25. Integrated Challenge (example with numbers 36 and 55)
