HCF is the largest number that divides all given numbers completely. LCM is the smallest number that is divisible by all given numbers. Think of HCF as the biggest common piece, and LCM as the smallest common whole
💡Key Properties
• For two numbers a and b: HCF × LCM = a × b
• HCF is always less than or equal to the smallest number
• LCM is always greater than or equal to the largest number
• If two numbers are co-prime (HCF = 1), then LCM = product of numbers
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Formula Block
Memorise — at least one formula appears in every paper
HCF × LCM = Product of two numbers
For three numbers: HCF(a,b,c) × LCM(a,b,c) ≠ a × b × c
HCF by Division Method: Keep dividing larger by smaller until remainder is zero
LCM = (a × b) ÷ HCF(a,b)
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Exam Patterns
What examiners ask — read before attempting PYQs
📋SSC CGL typically asks
direct HCF/LCM calculation, word problems involving bells/traffic lights, finding numbers when HCF/LCM is given, and ratio-based problems. Most questions are 2-mark difficulty level
⚡Powerful Shortcut - Product Rule
For any two numbers, if you know any three values among {number1, number2, HCF, LCM}, you can find the fourth instantly using: HCF × LCM = number1 × number2
Next time = 9:00 AM + 60 minutes = 10:00 AM
Time-Saving Trick - Division Method:
For HCF: Keep dividing larger number by smaller until remainder is zero. Last divisor is HCF.
48 ÷ 72: 72 = 48 × 1 + 24, then 48 = 24 × 2 + 0. HCF = 24
⚡Another Speed Trick - Co-prime Check
If two numbers have no common factors except 1, their HCF = 1 and LCM = their product
✏️Examples
(7,11), (15,16), (25,28)
⚠️#1 Most Common Mistake
Students confuse the formula application for three numbers. The golden rule HCF × LCM = product applies ONLY to two numbers. For three or more numbers, this formula does NOT work. Always work with pairs or use prime factorization method for multiple numbers.
This mistake costs students 2-4 marks per exam
💡Practical Tip
In word problems, LCM gives 'together again' scenarios (bells, traffic lights). HCF gives 'maximum equal groups' scenarios (distribution problems). Identify the scenario type first, then apply the appropriate concept.
Two bells ring at intervals of 18 minutes and 24 minutes respectively. If they ring together at 9:00 AM, at what time will they ring together again?
Practice 2easy
Find the LCM of 24, 36, and 48.
Practice 3easy
The HCF of two numbers is 15. If the numbers are 45 and 75, verify which statement is true.
Practice 4easy
Three ropes of lengths 48 m, 60 m, and 72 m are to be cut into equal pieces without any waste. What is the maximum length of each piece?
Practice 5medium
The HCF of three numbers is 6. If the numbers are in the ratio 2:3:4, and their LCM is 360, what is the largest number?
Practice 6medium
The HCF of two numbers is 15. If the numbers are in the ratio 3:5, what is the sum of the two numbers?
Practice 7medium
Two numbers have an HCF of 8 and an LCM of 240. How many pairs of such numbers are possible?
Practice 8medium
Three bells ring at intervals of 18, 24, and 36 minutes respectively. If they all ring together at 9:00 AM, at what time will they ring together again?
Practice 9hard
The HCF of two numbers is 16. Their LCM is 240. How many pairs of numbers satisfy this condition?
Practice 10hard
Two numbers have an HCF of 14 and an LCM of 420. If the sum of the two numbers is 154, find the difference between them.
Practice 11hard
Three numbers are in the ratio 2:3:4. Their LCM is 144. Find the HCF of the three numbers.
60-Second Revision — LCM and HCF
Formula: HCF × LCM = a × b (only for two numbers, not three)
Remember: Prime factorization uses lowest powers for HCF, highest for LCM
Trap: Three number product rule does NOT apply
Shortcut: Co-prime numbers have HCF = 1, LCM = their product