Class 7 | Ganita Prakash Part-II | Chapter 3
๐ Finding Common Ground
HCF ยท LCM ยท Euclid's Algorithm ยท Real-Life Applications
๐ 3.1 โ HCF (Highest Common Factor)
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๐ The Tile Story!
Sameeksha's room: 12 ft ร 16 ft. She wants the largest square tile that fits exactly.
Answer = HCF(12, 16) = 4 ft tiles!
Sameeksha's room: 12 ft ร 16 ft. She wants the largest square tile that fits exactly.
Answer = HCF(12, 16) = 4 ft tiles!
HCF = Greatest Common Divisor (GCD)
The largest number that divides ALL given numbers exactly.
HCF(12, 16):
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
Common factors: 1, 2, 4
HCF = 4 โ
The largest number that divides ALL given numbers exactly.
HCF(12, 16):
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
Common factors: 1, 2, 4
HCF = 4 โ
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๐ข Methods to Find HCF
Method 1: Prime Factorisation
24 = 2 ร 2 ร 2 ร 3 = 2ยณ ร 3
36 = 2 ร 2 ร 3 ร 3 = 2ยฒ ร 3ยฒ
HCF = product of lowest powers of common primes
HCF(24, 36) = 2ยฒ ร 3 = 12
24 = 2 ร 2 ร 2 ร 3 = 2ยณ ร 3
36 = 2 ร 2 ร 3 ร 3 = 2ยฒ ร 3ยฒ
HCF = product of lowest powers of common primes
HCF(24, 36) = 2ยฒ ร 3 = 12
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Method 2: Euclid's Division Algorithm (Faster!)
HCF(48, 18):
48 = 2 ร 18 + 12
18 = 1 ร 12 + 6
12 = 2 ร 6 + 0 โ remainder = 0!
HCF = 6 โ
Divide larger by smaller. Then divide previous divisor by remainder. Repeat until remainder = 0!
HCF(48, 18):
48 = 2 ร 18 + 12
18 = 1 ร 12 + 6
12 = 2 ร 6 + 0 โ remainder = 0!
HCF = 6 โ
Divide larger by smaller. Then divide previous divisor by remainder. Repeat until remainder = 0!
Real-life HCF problems:
- Largest equal groups from two quantities โ HCF
- Largest tile/square that fits a rectangle โ HCF of dimensions
- Maximum bags from two different quantities โ HCF
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๐ 3.2 โ LCM (Least Common Multiple)
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LCM = Least Common Multiple
The smallest number that is a multiple of ALL given numbers.
LCM(4, 6):
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
LCM = 12 โ
The smallest number that is a multiple of ALL given numbers.
LCM(4, 6):
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24...
LCM = 12 โ
LCM by Prime Factorisation:
LCM(12, 18):
12 = 2ยฒ ร 3 18 = 2 ร 3ยฒ
LCM = product of highest powers of all primes
LCM = 2ยฒ ร 3ยฒ = 4 ร 9 = 36
LCM(12, 18):
12 = 2ยฒ ร 3 18 = 2 ร 3ยฒ
LCM = product of highest powers of all primes
LCM = 2ยฒ ร 3ยฒ = 4 ร 9 = 36
๐ Real-life LCM!
Bus A comes every 12 min. Bus B comes every 18 min. Both are here now.
When will both be here at the same time again? โ LCM(12,18) = 36 minutes!
Bus A comes every 12 min. Bus B comes every 18 min. Both are here now.
When will both be here at the same time again? โ LCM(12,18) = 36 minutes!
- When do things repeat together โ LCM
- Smallest common denominator for fractions โ LCM
- "When will they meet again?" โ LCM
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๐ 3.3 โ The HCF ร LCM Relationship
๐ Beautiful Formula:
Example: HCF(12,18) = 6 LCM(12,18) = 36
6 ร 36 = 216 and 12 ร 18 = 216 โ
Very useful! If you know HCF and one number, find LCM:
LCM = (a ร b) รท HCF
HCF ร LCM = Product of the two numbers
Example: HCF(12,18) = 6 LCM(12,18) = 36
6 ร 36 = 216 and 12 ร 18 = 216 โ
Very useful! If you know HCF and one number, find LCM:
LCM = (a ร b) รท HCF
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HCF vs LCM โ Quick Comparison:
| HCF | LCM | |
|---|---|---|
| What | Largest common divisor | Smallest common multiple |
| Method (PF) | Lowest powers of common primes | Highest powers of all primes |
| Result | โค smallest number | โฅ largest number |
| Real use | Grouping, tiling, sharing | Scheduling, fractions, timing |
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๐ Quick Cheatsheet
HCF = Largest
common factor
โค smallest number
common factor
โค smallest number
LCM = Smallest
common multiple
โฅ largest number
common multiple
โฅ largest number
HCF ร LCM =
a ร b
(for 2 numbers)
a ร b
(for 2 numbers)
HCF: lowest powers
LCM: highest powers
(prime factors)
LCM: highest powers
(prime factors)
๐ Chapter 3 Complete! ๐
Made with โค๏ธ by @edugrown
Made with โค๏ธ by @edugrown