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In everyday life, we often encounter situations where we need to divide things into equal groups or find common timings for events. For example, if you have Rs.360 and Rs.480 and want to divide them into equal bundles without splitting any rupee notes, how many rupees should each bundle contain? Or, if two buses start from a station every 12 and 15 minutes respectively, when will they arrive together again?
These questions involve two important concepts in number theory: HCF (Highest Common Factor) and LCM (Least Common Multiple). Understanding these helps not only in solving such practical problems but also forms a foundation for many competitive exam questions.
In this chapter, we will explore what HCF and LCM mean, how to find them using different methods, their relationship, and how to apply them effectively.
The Highest Common Factor of two or more numbers is the largest number that divides all of them exactly, leaving no remainder.
For example, consider the numbers 36 and 48. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36, and the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The common factors are 1, 2, 3, 4, 6, 12. The highest among these is 12, so the HCF of 36 and 48 is 12.
There are several methods to find the HCF, including prime factorization and the Euclidean algorithm. Let's understand these methods step-by-step.
The Least Common Multiple of two or more numbers is the smallest number that is exactly divisible by all of them.
For example, consider the numbers 12 and 15. Multiples of 12 are 12, 24, 36, 48, 60, 72, ... and multiples of 15 are 15, 30, 45, 60, 75, ... The smallest common multiple is 60, so the LCM of 12 and 15 is 60.
Methods to find LCM include prime factorization, using the HCF, and listing multiples. Prime factorization is often the most reliable method for larger numbers.
There is a very important and useful relationship between the HCF and LCM of two numbers \(a\) and \(b\):
This formula is very helpful because if you know any two of the three values (HCF, LCM, or the numbers), you can find the third easily.
| Numbers (a, b) | HCF(a,b) | LCM(a,b) | Product (a x b) | HCF x LCM |
|---|---|---|---|---|
| 8, 12 | 4 | 24 | 96 | 4 x 24 = 96 |
| 15, 20 | 5 | 60 | 300 | 5 x 60 = 300 |
| 21, 14 | 7 | 42 | 294 | 7 x 42 = 294 |
Step 1: Find the prime factors of each number.
36 = 2 x 2 x 3 x 3 = \(2^2 \times 3^2\)
48 = 2 x 2 x 2 x 2 x 3 = \(2^4 \times 3^1\)
Step 2: Identify the common prime factors with the smallest powers.
Common prime factors are 2 and 3.
Smallest power of 2 is \(2^2\), smallest power of 3 is \(3^1\).
Step 3: Multiply these to get the HCF.
HCF = \(2^2 \times 3^1 = 4 \times 3 = 12\)
Answer: The HCF of 36 and 48 is 12.
Step 1: Find the prime factors of each number.
12 = 2 x 2 x 3 = \(2^2 \times 3^1\)
15 = 3 x 5 = \(3^1 \times 5^1\)
Step 2: Take all prime factors with the highest powers.
Prime factors: 2, 3, 5
Highest powers: \(2^2\), \(3^1\), \(5^1\)
Step 3: Multiply these to get the LCM.
LCM = \(2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\)
Answer: The LCM of 12 and 15 is 60.
The Euclidean algorithm finds the HCF by repeatedly replacing the larger number by the remainder when divided by the smaller number until the remainder is zero.
Step 1: Divide 1071 by 462 and find the remainder.
1071 / 462 = 2 remainder 147 (since 462 x 2 = 924, 1071 - 924 = 147)
Step 2: Replace 1071 with 462, and 462 with 147.
Now find HCF(462, 147)
Step 3: Divide 462 by 147.
462 / 147 = 3 remainder 21 (147 x 3 = 441, 462 - 441 = 21)
Step 4: Replace 462 with 147, and 147 with 21.
Now find HCF(147, 21)
Step 5: Divide 147 by 21.
147 / 21 = 7 remainder 0
Since remainder is zero, the HCF is the divisor at this step, which is 21.
Answer: The HCF of 1071 and 462 is 21.
This problem asks for the time interval after which both events coincide again. This is the LCM of 12 and 15.
Step 1: Find the prime factors:
12 = \(2^2 \times 3\)
15 = \(3 \times 5\)
Step 2: Take all prime factors with highest powers:
LCM = \(2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60\)
Answer: Both events will occur together after 60 days.
We use the relationship:
Step 1: Substitute the known values.
6 x LCM = 2160
Step 2: Solve for LCM.
LCM = \(\frac{2160}{6} = 360\)
Answer: The LCM of the two numbers is 360.
When to use: When numbers are large or prime factorization is time-consuming.
When to use: When either HCF or LCM is missing in a problem.
When to use: When numbers are less than 20 and factorization is not preferred.
When to use: To quickly identify common factors.
When to use: When performing prime factorization manually.
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