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Imagine you have two piles of toys and you want to arrange them into equal groups without any toys left over. Or maybe you need to find when two different bus routes will meet at the same stop again. This is where HCF (Highest Common Factor) and LCM (Least Common Multiple) come into play! These two buddies help us find common patterns and shared amounts between numbers, making many real-life problems super easy to solve.
When you have two numbers, say A and B, and you need to find their HCF quickly, first check if the smaller number divides the larger number. If it does, the smaller number itself is the HCF! If not, check if half of the smaller number divides both. Keep trying divisors of the smaller number from largest to smallest.
For LCM of two numbers, say A and B, often you can start with the larger number and check its multiples. The first multiple of the larger number that is also perfectly divisible by the smaller number is your LCM. This is super fast when numbers are small or one is a multiple of the other!
Remember the golden rule: Number 1 × Number 2 = HCF × LCM. If you know any three of these four values, you can instantly find the fourth. This trick is a huge time-saver in exams where these types of questions are common.
When a problem asks for the largest number that divides X, Y, Z leaving remainders R1, R2, R3, you are essentially looking for the HCF of (X-R1), (Y-R2), and (Z-R3). This simplifies the problem from remainders to simple HCF.
If two numbers have no common factors other than 1 (they are called 'co-prime' numbers, like 7 and 11), then their HCF is always 1. And their LCM is simply the product of the two numbers. This is a quick mental math hack!
Let's start with HCF, which stands for Highest Common Factor. Think of it like this: A factor is a number that divides another number exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 18 are 1, 2, 3, 6, 9, 18. When we look at both lists, we see some numbers that are common to both. These are 1, 2, 3, 6. Among these common factors, the largest one is 6. So, the HCF of 12 and 18 is 6.
HCF is also sometimes called the Greatest Common Divisor (GCD). It helps us find the biggest group size we can make when dividing items equally from different sets.
Now let's talk about LCM, which means Least Common Multiple. A multiple of a number is what you get when you multiply that number by other whole numbers. For example, multiples of 4 are 4, 8, 12, 16, 20, 24, .... Multiples of 6 are 6, 12, 18, 24, 30, .... Looking at both lists, common multiples are 12, 24, .... The smallest among these common multiples is 12. So, the LCM of 4 and 6 is 12.
LCM helps us find the smallest quantity that can be divided perfectly by two or more numbers. It's useful when you need things to 'meet up' again after different cycles, like bus schedules or flashing lights.
Remember the golden rule for any two numbers 'a' and 'b': Product of numbers = HCF(a, b) × LCM(a, b). This rule is super important for competitive exams!
Product of two numbers
Number 1 × Number 2 = HCF(Number 1, Number 2) × LCM(Number 1, Number 2)HCF of Fractions
HCF(a/b, c/d) = HCF(a, c) / LCM(b, d)LCM of Fractions
LCM(a/b, c/d) = LCM(a, c) / HCF(b, d)HCF of Numbers with Powers
HCF of (2^a * 3^b, 2^c * 3^d) = 2^min(a,c) * 3^min(b,d)LCM of Numbers with Powers
LCM of (2^a * 3^b, 2^c * 3^d) = 2^max(a,c) * 3^max(b,d)| Feature | HCF (Highest Common Factor) | LCM (Least Common Multiple) |
|---|---|---|
| What it finds | Largest number that divides given numbers exactly | Smallest number that is exactly divisible by given numbers |
| What it's also called | Greatest Common Divisor (GCD) | Lowest Common Multiple |
| Relation to numbers | HCF always divides the given numbers | LCM is always divisible by the given numbers |
| Prime Factorization | Product of common prime factors with lowest powers | Product of all prime factors with highest powers |
| Typical Use | Distributing items into largest equal groups, tiling | Scheduling events, finding common intervals, smallest quantity |
Q: Find the HCF of 24 and 36.
Q: Find the LCM of 15 and 25.
Q: The product of two numbers is 1800. If their HCF is 15, what is their LCM?
Q: Three bells ring at intervals of 10, 12, and 15 minutes respectively. If they all ring together at 10:00 AM, at what time will they next ring together?
Two friends cycle on a circular track. One takes 12 minutes per round and the other takes 18 minutes. If they start at the same time and spot, when will they meet at the starting point again?
You have 30 ladoos and 45 jalebis. You want to pack them into identical boxes, with each box having the same number of ladoos and jalebis, without mixing different sweets. What's the maximum number of such boxes you can make?
Three traffic lights at different crossings change after every 40 seconds, 60 seconds, and 90 seconds. If they all change simultaneously at 8:00 AM, when will they change simultaneously again?
A rectangular floor is 8 meters 50 cm long and 6 meters 25 cm wide. You want to cover it with square tiles of the largest possible size without cutting any tiles. What is the side length of the largest square tile you can use?
The HCF of two numbers is 8. Which of the following CANNOT be their LCM?
Two numbers are in the ratio 3:4. If their HCF is 7, what are the numbers?
Find the greatest number which divides 60, 90, and 120 exactly.
The LCM of two numbers is 200 and their HCF is 10. If one number is 50, what is the other number?
1What is the HCF of 18, 24, and 30?
2Find the LCM of 8, 12, and 16.
3The HCF of two numbers is 6 and their LCM is 72. If one number is 18, find the other number.
4What is the HCF of (2/3), (4/9), and (5/6)?
5What is the LCM of (3/4), (5/6), and (6/7)?
6What is the greatest number that divides 102 and 170 exactly?
7Find the smallest number which is exactly divisible by 15, 20, and 25.
8The ratio of two numbers is 2:3 and their LCM is 48. What is their HCF?
9What is the HCF of 105, 120, and 150?
10The product of two numbers is 675. If their HCF is 15, what is their LCM?
11Find the least number which when divided by 12, 15, and 20 leaves no remainder in each case.
When you have two numbers, say A and B, and you need to find their HCF quickly, first check if the smaller number divides the larger number. If it does, the smaller number itself is the HCF! If not, check if half of the smaller number divides both. Keep trying divisors of the smaller number from largest to smallest.
For LCM of two numbers, say A and B, often you can start with the larger number and check its multiples. The first multiple of the larger number that is also perfectly divisible by the smaller number is your LCM. This is super fast when numbers are small or one is a multiple of the other!
Remember the golden rule: Number 1 × Number 2 = HCF × LCM. If you know any three of these four values, you can instantly find the fourth. This trick is a huge time-saver in exams where these types of questions are common.
When a problem asks for the largest number that divides X, Y, Z leaving remainders R1, R2, R3, you are essentially looking for the HCF of (X-R1), (Y-R2), and (Z-R3). This simplifies the problem from remainders to simple HCF.
If two numbers have no common factors other than 1 (they are called 'co-prime' numbers, like 7 and 11), then their HCF is always 1. And their LCM is simply the product of the two numbers. This is a quick mental math hack!
Number 1 × Number 2 = HCF(Number 1, Number 2) × LCM(Number 1, Number 2)HCF(a/b, c/d) = HCF(a, c) / LCM(b, d)LCM(a/b, c/d) = LCM(a, c) / HCF(b, d)+2 more formulas below