HCF (Highest Common Factor) is the greatest number that divides two or more numbers without leaving any remainder.
It is also called:
HCF is widely used in:
Method 1: Prime Factorization
Example:
HCF(36, 60)
36 = 2² × 3²
60 = 2² × 3 × 5
Common factors = 2² × 3 = 12
Method 2: Division Method (Euclid’s Method)
Use repeated division:
HCF(a, b) = HCF(b, remainder)
Continue until remainder becomes 0.
The last divisor is the HCF.
Method 3: Listing Common Factors
⭐ 5 Solved Examples of HCF (With Explanation)
24 = 2³ × 3
36 = 2² × 3²
Common = 2² × 3 = 12
HCF = 12
18 = 2 × 3²
27 = 3³
45 = 3² × 5
Common = 3² = 9
HCF = 9
120 ÷ 72 = 48
72 ÷ 48 = 24
48 ÷ 24 = 0
HCF = 24
14 = 2 × 7
35 = 5 × 7
Common = 7
HCF = 7
48 = 2⁴ × 3
64 = 2⁶
80 = 2⁴ × 5
Common = 2⁴ = 16
HCF = 16
📝Explanation: Prime factors:
8 = 2³
12 = 2² × 3
Common factor is 2² = 4.
Thus HCF = 4 because this is the largest number that divides both 8 and 12 without leaving any remainder.
📝 Description:
9 = 3²
27 = 3³
Common = 3² = 9.
9 divides both numbers exactly, while no larger number does. Hence the correct HCF is 9.
📝 Description: Prime factors:
16 = 2⁴
20 = 2² × 5
Common = 2² = 4.
Thus the largest common divisor of both numbers is 4.
📝 Description:
15 = 3 × 5
25 = 5 × 5
Common = 5.
5 is the greatest number dividing both without remainder. Hence HCF = 5.
📝 Description:
512 = 2² × 3
18 = 2 × 3²
30 = 2 × 3 × 5
Common = 2 × 3 = 6
Thus HCF = 6, the largest number dividing all three.
📝 Description:
11 = 11
121 = 11²
Common = 11
So the highest factor common to both is 11.
📝 Description:
40 = 2³ × 5
56 = 2³ × 7
Common = 2³ = 8
Thus HCF = 8.
📝 Description:
48 = 2⁴ × 3
180 = 2² × 3² × 5
Common = 2² × 3 = 12
So HCF = 12.
📝 Description: 32 = 2⁵
96 = 2⁵ × 3
160 = 2⁵ × 5
Common factor = 2⁵ = 32
But all three share only 2⁴ = 16.
Thus HCF = 16.
📝 Description: Prime numbers have no common factors except 1, unless they are the same prime. Therefore, the HCF of any two different primes is always 1.
📝 Description:
81 = 3⁴
108 = 2² × 3³
Common = 3³ = 27
Thus HCF = 27.
📝 Description:
16 = 2⁴
27 = 3³
No common prime factors → co-primes.
Hence HCF = 1.
📝 Description:
50 = 2 × 5²
75 = 3 × 5²
Common = 5² = 25
Thus HCF = 25.
📝 Description:
144 ÷ 96 = 48
96 ÷ 48 = 0
Last non-zero divisor = 48
Thus HCF = 48.
📝 Description:
HCF(7, 49) = 7
Other pairs have HCF > 7.
Thus the only correct pair is 7 and 49.
📝 Description:
72 = 2³ × 3²
108 = 2² × 3³
180 = 2² × 3² × 5
Common = 2² × 3² = 4 × 9 = 36?
But 36 doesn’t divide 72 evenly? Yes it does.
Actually dividing:
72 ÷ 36 = 2
108 ÷ 36 = 3
180 ÷ 36 = 5
So HCF = 36
📝 Description: Both are prime numbers. Different primes have no common factors except 1. Hence HCF = 1.
📝 Description:
120 = 2³ × 3 × 5
150 = 2 × 3 × 5²
Common = 2 × 3 × 5 = 30
But 30 must divide 120 and 150 → Yes.
Thus HCF = 30.
📝 Description: Any 3 consecutive integers always have different prime factor structures. They never share any common factor except 1. Thus their HCF is always 1.
📝 Description:
84 = 2² × 3 × 7
140 = 2² × 5 × 7
196 = 2² × 7²
Common = 2² × 7 = 28
Thus HCF = 28.
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