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On this page you will get the definition, methods to find HCF and formulas of HCF that are helpful in the preparation of HCF questions. HCF stands for Highest Common Factor. It is highest positive integer that divides two or more integers. That’s why it is also known as Greatest Common Divisor (GCD).
There are multiple ways to find the highest common factor or HCF of the given numbers. Irrespective of the method, the answer to the HCF of the numbers would always be the same. Here are 3 methods to calculate the HCF of two numbers:
Question : Find the Highest Common Factor of 55, 65 and 75.
A. 2B. 3C. 4 D. 5
Solution:Given, three numbers as 55, 65 and 75.
We know, 55 = 5 × 11
65 = 5 × 13
75 = 5 × 15
From the above expression, we can say 5 is the only common factor for all three numbers.
Therefore, 5 is the HCF of 55, 65 and 75.
Question : 6 and 7 are two co-prime numbers. Using these numbers verify the HCF formula, LCM of Co-prime Numbers = Product of the numbers.
A. 45
B. 46
C. 42
D. 47
Solution: Let’s find the LCM and HCF of 6 and 7
6 = 1×2×3
7= 1×7
LCM = 1×2×3×7 = 42 HCF = 1
The HCF formula states that LCM of Co-prime Numbers = Product of the numbers
Product of the numbers = 6×7 = 42
Hence, LCM of Co-prime Numbers = Product of the numbers i.e. 42 = 42. Therefore, it is verified.
Question : Find the HCF of 18/10, 12/15, 24/20, 30/5
A. 3/400
B. 4/500
C. 6/400
D. 8/400
Solution: According to the HCF formula, HCF = HCF of numerator/LCM of the denominator.
So finding the HCF of the numerators:
18 = 2×3×3
12 = 2×2×3
24 = 2×2×2×3
30 = 2×5 × 3
Therefore, the HCF of 9, 8, 12, 18 = 3
10 = 1×2×5
15 = 1×3×5
20 = 1×2×4×5
5 = 1×5
Therefore, the LCM of 10, 15, 20, 5 = 2×5×2×4×5 = 400
HCF = HCF of numerators/LCM of the denominators
HCF = 3/400
Therefore, the HCF of 9/10, 6/15, 12/20, 18/5 is 3/400.
Question : Rakesh joshi has calculated the H.C.F. and L.C.M. of two 2- digit numbers as16 and 480 respectively. The numbers are :
A. 40, 48 B. 60, 72 C. 64, 80D. 80, 96
Ans: 80, 96H.C.F. of the two 2-digit numbers= 16Hence, the numbers can be expressed as 16x and 16y, where x and y are prime to each other.Now,First number \times second number = H.C.F. \times L.C.M. => 16x \times 16y = 16 \times 480 => xy = 16 \times \frac{480}{16} \times 16= 30 ´ The possible pairs of x and y, satisfying the condition xy = 30 are :(3, 10), (5, 6), (1, 30), (2, 15)So the numbers are (48,160) (80,96)
Question : The HCF of the two numbers is 29 & their sum is 174. What are the numbers?
A. 29 and 145B. 30 and 150C. 40 and 140D. 65 and 130
Solution:
Let the two numbers be 29x and 29y.
Given, 29x + 29y = 174
29(x + y) = 174
x + y = 174/29 = 6
Since x and y are co-primes, therefore, possible combinations would be (1,5), (2,4), (3,3).
The only combination that follows the co-prime part is (1,5)
For (1,5): 29 x = 29 x 1 and 29 y = 29 (5) = 145
Therefore, the required numbers are 29 and 145.
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