HCF and LCM: Shortcuts & Tricks | Quantitative Techniques for CLAT PDF Download

HCF (Highest Common Factor)

The highest common factor (HCF) or greatest common measure (GCM) or Greatest common divisor (GCD) is the largest number that divides two or more given numbers.

Methods to find HCF 

1. Division method: This method uses successive division to find the HCF. To find the HCF of two numbers N1 and N2, the smaller of the two, say N1, divides N2. The remainder R1 then divides N1 followed by the remainder R2 dividing R1, and so on till a remainder of 0 is reached. The last divisor is HCF.
2. Prime Factorization method: Express each one of the given numbers as a product of prime numbers. The product of the least powers of common prime factor gives HCF. For example,

The HCF of 96, 144, and 240 can be found by the prime factorization of these numbers.
96 = 25×31
144 = 24×32
240 = 24×31×51
HCF = 24×31 = 48

Tricks to Find HCF easily

Q1: Find HCF of 12 and 16.
Sol: Find the difference between 12 and 16. The difference is 4. Now, check whether the numbers are divisible by the difference. 12 is divisible by 4 and 16 is divisible by 4.
Hence, the HCF is 4.

Q2: Find HCF of 18 and 22.
Sol: Find the difference between 18 and 22. The difference is 4. Now, check whether the numbers are divisible by the difference. Both 18 and 22 are not divisible by 4. So take the factors of the difference. The factors of 4 are 2*2*1. Now, check whether the numbers are divisible by the factors. 18 and 22 are divisible by factor 2.
Hence, the HCF is 2.

Note: If there are more than two numbers, take the least difference.

Q3: Find HCF of 10, 14, and 34.
Sol: Find the difference between 10 and 14. The difference is 4. Now find the difference between 14 and 34. The difference is 20. Now find the difference between 10 and 34. The difference is 24. Out of 4, 20, and 24, 4 is the least difference. So, we take the number 4 and check whether the numbers are divisible. 10,14, and 34 are not divisible by 4. So take the factors of the difference. The factors of 4 are 2*2*1. Now, check whether the numbers are divisible by the factors. 10,14 and 34 are divisible by factor 2.

Q4: Find HCF of 15, 25, and 35.
Sol: Find the difference between 15, 25, and 35. The least difference is 10. But 10 is not divisible by 15, 25, and 35. So, we take the factors of the difference. The factors of 10 are 5*2*1. Now, check whether the numbers are divisible by the factors. 15, 25, and 35 are divisible by 5.

LCM (Least Common Multiple)

The least number which is exactly divisible by each one of the given numbers is called their LCM.

Methods to find LCM 

Prime Factorization method:  After performing prime factorization of numbers, all prime numbers in their highest index are selected and multiplied to get the LCM. For example,
The LCM of 96, 144, and 240 can be found by the prime factorization of these numbers.
96 = 25×31
144 = 24×32
240 = 24×31×51
LCM = 25×32×51 = 1440

Tricks to Find LCM easily

Q1: Find LCM of 2,4,8,16.
Sol: Choose the largest number. In this example, the largest number is 16. Check whether 16 is divisible by all other remaining numbers. 16 is divisible by 2, 4, 8. Hence, the LCM is 16.

Q2: Find the LCM of 2,3,7,21.
Sol: Choose the largest number. The largest number is 21. Check whether 21 is divisible by all other remaining numbers. 21 is divisible by 3 and 7 but not by 2. So multiply 21 and 2. The result is 42. Now, check whether 42 is divisible by 2, 3, 7. Yes, 42 is divisible. Hence, the LCM is 42. 

Q3: Find the LCM of 3,5,15,30
Sol: Choose the largest number. The largest number is 30. Check whether 30 is divisible by all other remaining numbers. 30 is divisible by 3, 5, and 15. Hence, the LCM is 30.

Q4: Find the LCM of 8,24,48,96
Sol: Choose the largest number. The largest number is 96. Check whether 96 is divisible by all other remaining numbers. 96 is divisible by 8, 24, and 48. Hence, the LCM is 96.

Q 5: Find the LCM of 12,36,60,108
Sol: Choose the largest number. The largest number is 108. Check whether 108 is divisible by all other remaining numbers. 108 is divisible by 12, 36, and 60. Hence, the LCM is 108.