LCM & HCF
Aptitude H.C.F and L.C.M Concepts and Formulas
Points to remember:
1. Factor: Factor of a number is a number which divides the number evenly. For example, 15 can be divided evenly by 1, 3, 5, and 15 so the factors of 15 are 1, 3, 5 and 15.
2. Common Factor: A common factor is a number which divides the two or more numbers exactly or evenly.
3. Highest Common Factor (H.C.F): HCF of the two numbers is the highest number of the two numbers which divides them exactly.
4. Methods to find the H.C.F of two or more numbers:
- Method of prime factors: Break the given numbers into prime factors and then find the product of all prime factors common to all the numbers. The product will be the required HCF.
- Method of Division: Divide the greater number by smaller number, divide the divisor by the remainder, and divide the remainder by the next remainder, and so on until no number is left. The last divisor is the required HCF.
- To find the HCF of more than two numbers: Firstly find the HCF of first two numbers and then find the HCF of this HCF and the third number and so on and the last HCF will be the required HCF.
- HCF of decimals: Firstly, make the same number of decimal places in all the given numbers and then find the HCF of the numbers as if they are integers and mark the decimal places as there in each of the numbers.
- HCF of the vulgar fraction: HCF of two or more fraction is the highest fraction which is exactly divisible by each of the fractions.
5. Common Multiple: Common multiple is the common number between two or more numbers which is exactly divisible by each number.
6. Least Common Multiple (LCM): LCM is the smallest number between the two numbers which exactly divides both of them.
7. Methods to find the LCM of two or more given numbers:
- Method of Prime Factors: Resolve the given numbers into their prime factors and then find the product of the highest power of all the factors that occur in the given numbers. This product is the LCM.
- The LCM of several small numbers can be easily found by the following rule:
Write down the given numbers in a line separating them by commas. Divide them by any one of the prime number 2, 3, 5, 7, etc., which exactly divides at least any two of the given numbers. Write down the quotient and the undivided numbers in a line below the first. Repeat the process to get a line of numbers which are Prime to one another. The product of all divisors and the numbers in the last line will be the required LCM.
- LCM of decimals: First make the same number of decimal places in all the given numbers. Then find their LCM and mark in the result as many decimal places as there are in each of the numbers.
- LCM of fractions: The LCM of two or more fractions is the least fraction or integer which is exactly divisible by each of them.
Express the fractions in their lowest terms, then
8. HCF x LCM = 1st number x 2nd number
Practice Question :
1) What is the HCF of 1095 and 1168?
2) Find the HCF of 210, 385, and 735.
4) What will be the HCF of 608, 544; 638, 783; and 425, 476 respectively?
- 32, 29, 17
- 17, 32, 29
- 29, 32, 17
- 32, 17, 29
6) If the HCF of two numbers is 27, and their sum is 216, find these numbers.
- 27, 189
- 154, 162
- 108, 108
- 81, 189
7) Two numbers are in the ratio of 15:11. If the HCF of numbers is 13, find the numbers.
- 75, 55
- 105, 77
- 15, 11
- 195, 143
8) Find the greatest integer that divides 358, 376, and 334 and leaves the same remainder in each case.
9) Three bells toll at intervals of 36 sec, 40 sec, and 48 sec respectively. They start singing together at a particular time. When will they toll next together?
- 6 minutes
- 12 minutes
- 18 minutes
- 24 minutes
10) The LCM of two numbers is 7700, and their HCF is 11. If one of these numbers is 275, what is the other number?
11) The LCM of two numbers is 8919, and their sum is 1000. The two numbers are:
- 993 and 7
- 989 and 11
- 991 and 9
- 987 and 13
12) If the LCM of two numbers is 70 and their HCF is 2, find the numbers.
- 2, 35
- 6, 70
- 4, 70
- 14, 10
13) Three different containers contain 496 liters, 403 liters, and 713 liters of water, respectively. What is the capacity of the biggest container that can measure three different quantities evenly?
- 1 liter
- 7 liters
- 31 liters
- 41 liters
14) The least number which when divided by 2, 3, 4, 5 and 6 leaves the same remainder 1 in each case. If the same number is divided by 7, it leaves no remainder. Find the number.
15) Which is the greatest three-digit number which when divided by 6, 9 and 12 leaves a remainder of 3 in each case?
16) There are two numbers. HCF of both the numbers is 11, and their LCM is 693. If the first number is 77, find the second number?
17) Find the greatest number which divides 34, 90, and 104 and leaves the same remainder in each case.
18) What is the least multiple of 7 which when divided by 6, 9, 15, and 18 respectively, leaves 4 as the remainder in each case?
19) Which least number should be subtracted from 1936 so that the resulting number, when divided by 9, 10, and 15, will leave the same remainder 8 in each case?
20) Find the least number which when divided by 5, 6, 7, and 8 leaves remainder 3, but when divided by 9 leaves remainder 0.