K2KARAKORAM
मराठीEnglish
Open the app

State Board · Class 6 · Mathematics

Lesson 5: GCD and LCM

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Divisors, multiples and prime factors

From daily life

Teacher has 20 chocolates and wants to make equal groups with none left over. Groups of 1, 2, 4, 5, 10 or 20 work. These numbers are the divisors of 20.

Simple definition

A divisor (भाजक / factor) divides a number exactly. The number is then a multiple (गुणाकार / multiple) of that divisor. A prime factor (मूळ अवयव) is a divisor that is also a prime number.

Rule 1: Divisors and multiples — Divisors of 28: 1, 2, 4, 7, 14, 28. So 28 is a multiple of each of them. Multiples of 7: 7, 14, 21, 28, … A prime number such as 53 has only two divisors, 1 and itself. The number 1 is neither prime nor composite.

  • Easy : Easy: Divisors of 10 are 1, 2, 5, 10.
  • Medium : Medium: Divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
  • Exam-type : Exam: Is 91 prime? No, 91 = 7 × 13, so it has more than two divisors.

Rule 2: Prime factors and the factor tree — Break a number into smaller factors until only primes remain: 90 = 9 × 10 = (3 × 3) × (2 × 5), so 90 = 2 × 3 × 3 × 5.

Factor tree of 60: keep splitting until only primes are left.602302153560 = 2 × 2 × 3 × 5Green = prime factors

In the figure : In the figure the factor tree of 60: 2 × 2 × 3 × 5.

  • Easy : Easy: 12 = 2 × 2 × 3.
  • Medium : Medium: 84 = 2 × 42 = 2 × 2 × 21 = 2 × 2 × 3 × 7.
  • Exam-type : Exam: 150 = 2 × 75 = 2 × 3 × 25 = 2 × 3 × 5 × 5.
Remember

Divisor divides exactly. Multiple is divisor × something. Prime has exactly two divisors.

Common mistake

Calling 1 a prime number. It has only one divisor.

Shortcut

Memory clue: "मूळ संख्या = फक्त दोन भाजक: 1 आणि स्वतः".

Spot the difference

TermExample
Divisors of 281, 2, 4, 7, 14, 28
Multiples of 77, 14, 21, 28
Prime factors of 902 × 3 × 3 × 5

Practice questions

Easy
  1. 1Write all divisors of 15.
  2. 2Write the first four multiples of 9.
  3. 3Is 29 a prime number?
Medium
  1. 4Is 57 a prime number?
  2. 5Write the prime factors of 48.
  3. 6Write the prime factors of 100.
Challenging
  1. 745 sweets are shared in equal groups with none left. What group sizes are possible?
  2. 8Why is 1 not a prime number?
Show answers (try them yourself first)
  1. 11, 3, 5, 15.
  2. 29, 18, 27, 36.
  3. 3Yes, only 1 and 29 divide it.
  4. 4No, 57 = 3 × 19.
  5. 52 × 2 × 2 × 2 × 3.
  6. 62 × 2 × 5 × 5.
  7. 7The divisors of 45: 1, 3, 5, 9, 15, 45.
  8. 8A prime number must have exactly two divisors, but 1 has only one.

Greatest common divisor (GCD)

From daily life

A tailor wants to cut two ribbons of 20 m and 30 m into equal pieces with no waste, and he wants the pieces as long as possible. He needs the greatest common divisor.

Simple definition

The GCD (मसावि; also called HCF) of two numbers is the greatest number that divides both exactly.

Rule 1: Listing method — Divisors of 20: 1, 2, 4, 5, 10, 20. Divisors of 30: 1, 2, 3, 5, 6, 10, 15, 30. Common divisors: 1, 2, 5, 10. The GCD is 10, so the tailor makes pieces of 10 m.

Divisors of 12 and 16 overlap in 1, 2 and 4; the greatest is 4.Divisors of 12Divisors of 163612816124Common: 1, 2, 4 GCD = 4

In the figure : In the figure the divisors of 12 and 16 overlap in 1, 2, 4.

  • Easy : Easy: GCD of 6 and 9: common divisors are 1, 3, so the GCD is 3.
  • Medium : Medium: GCD of 14 and 21: divisors 1, 2, 7, 14 and 1, 3, 7, 21; the GCD is 7.
  • Exam-type : Exam: GCD of 8 and 15 is 1; such numbers are called co-prime (सहमूळ).

Rule 2: Prime factor method — Write each number as a product of primes and multiply the common prime factors. 40 = 2 × 2 × 2 × 5 and 60 = 2 × 2 × 3 × 5 share 2 × 2 × 5, so the GCD is 20. Two different prime numbers always have GCD 1.

  • Easy : Easy: 18 = 2 × 3 × 3 and 12 = 2 × 2 × 3 share 2 × 3, so GCD = 6.
  • Medium : Medium: 54 = 2 × 3 × 3 × 3 and 72 = 2 × 2 × 2 × 3 × 3 share 2 × 3 × 3, so GCD = 18.
  • Exam-type : Exam: GCD of 24, 36 and 60: 24 = 2×2×2×3, 36 = 2×2×3×3, 60 = 2×2×3×5 share 2×2×3, so GCD = 12.
Live figure

Greatest common divisor (GCD)

In the app you can move this figure with your finger.

Open in the app
Remember

GCD = greatest common divisor. Co-prime numbers have GCD 1.

Common mistake

Taking the common divisor that is first in the list, not the greatest.

Shortcut

Memory clue: "मसावि = सामाईक भाजकांतला सर्वांत मोठा".

Spot the difference

MethodExample (20, 30)
Listing divisorsCommon: 1, 2, 5, 10 → 10
Prime factors2×2×5 and 2×3×5 share 2×5 = 10

Practice questions

Easy
  1. 1Find the GCD of 8 and 12.
  2. 2Find the GCD of 15 and 25.
  3. 3Find the GCD of 9 and 28.
Medium
  1. 4Find the GCD of 36 and 48.
  2. 5Find the GCD of 45 and 60.
  3. 6What is the GCD of the prime numbers 7 and 11?
Challenging
  1. 7Use the prime factor method: GCD of 56 and 84.
  2. 8Ropes of 24 m and 40 m are cut into the longest equal pieces. What is the length of a piece?
Show answers (try them yourself first)
  1. 14.
  2. 25.
  3. 31 (co-prime).
  4. 412.
  5. 515.
  6. 61.
  7. 756 = 2×2×2×7, 84 = 2×2×3×7; common 2×2×7 = 28.
  8. 824 = 2×2×2×3 and 40 = 2×2×2×5, so GCD = 8 m.

GCD by the division method

From daily life

For big numbers like 135 and 105, listing all divisors takes long. A quick method by repeated division finds the GCD fast.

Simple definition

In the division method (भागाकार पद्धत), divide the bigger number by the smaller, then the divisor by the remainder, and repeat until the remainder is 0. The last divisor is the GCD.

Rule 1: Division method — GCD of 72 and 30: 72 = 2 × 30 + 12; 30 = 2 × 12 + 6; 12 = 2 × 6 + 0. The last divisor is 6, so the GCD is 6.

  • Easy : Easy: 20 and 8: 20 = 2 × 8 + 4; 8 = 2 × 4 + 0. GCD = 4.
  • Medium : Medium: 84 and 36: 84 = 2 × 36 + 12; 36 = 3 × 12 + 0. GCD = 12.
  • Exam-type : Exam: 135 and 105: 135 = 1 × 105 + 30; 105 = 3 × 30 + 15; 30 = 2 × 15 + 0. GCD = 15.

Rule 2: Where we use the GCD — Use the GCD for the greatest equal size. The biggest square tile for a 20 m by 12 m room has side 4 m (GCD of 20 and 12). The longest equal pieces from 60 m and 100 m of cloth are 20 m long.

A 15 m by 9 m room covered exactly by 3 m square tiles.15 m9 m3 mGCD of 15 and 9 = 3Square tile side 3 m5 × 3 = 15 tilesLargest square tile that fits with no cutting

In the figure : In the figure a 15 m by 9 m room and 3 m square tiles.

  • Easy : Easy: Biggest square tile for an 8 m by 6 m room: GCD = 2 m.
  • Medium : Medium: A 20 m by 12 m room with 4 m tiles needs (20 ÷ 4) × (12 ÷ 4) = 5 × 3 = 15 tiles.
  • Exam-type : Exam: 60 m and 100 m cloth in 20 m pieces gives 60 ÷ 20 + 100 ÷ 20 = 3 + 5 = 8 pieces.
Live figure

GCD by the division method

In the app you can move this figure with your finger.

Open in the app
Remember

Last divisor with remainder 0 is the GCD. Check: it divides both numbers exactly.

Common mistake

Taking the last remainder (0) as the answer. The GCD is the last divisor.

Shortcut

Memory clue: "बाकी शून्य, भाजक तोच मसावि".

Spot the difference

StepAction
1Divide bigger by smaller
2Divide divisor by remainder
3Stop when remainder is 0

Practice questions

Easy
  1. 1Use the division method for 40 and 16.
  2. 2GCD of 45 and 30 by the division method.
  3. 3GCD of 56 and 24 by the division method.
Medium
  1. 4GCD of 91 and 35 by the division method.
  2. 5GCD of 17 and 5 by the division method.
  3. 6Find the side of the biggest square tile for a 16 m by 24 m room.
Challenging
  1. 7From 48 m and 72 m of rope, what is the longest equal piece?
  2. 8A student says the GCD of 54 and 36 is 12. How can you check, and what is the right answer?
Show answers (try them yourself first)
  1. 140 = 2×16 + 8; 16 = 2×8 + 0. GCD = 8.
  2. 245 = 1×30 + 15; 30 = 2×15 + 0. GCD = 15.
  3. 356 = 2×24 + 8; 24 = 3×8 + 0. GCD = 8.
  4. 491 = 2×35 + 21; 35 = 1×21 + 14; 21 = 1×14 + 7; 14 = 2×7 + 0. GCD = 7.
  5. 517 = 3×5 + 2; 5 = 2×2 + 1; 2 = 2×1 + 0. GCD = 1.
  6. 6GCD of 16 and 24 is 8 m.
  7. 7GCD of 48 and 72 is 24 m.
  8. 854 ÷ 12 leaves a remainder, so 12 is wrong. 54 = 1×36 + 18; 36 = 2×18 + 0, so the GCD is 18.

Least common multiple (LCM)

From daily life

Two lights blink every 4 seconds and every 6 seconds. They blink together first at 12 seconds. This is the idea of the least common multiple.

Simple definition

The LCM (लसावि) of given numbers is the smallest number that all of them divide exactly.

Rule 1: Listing multiples — Multiples of 9: 9, 18, 27, 36, … Multiples of 12: 12, 24, 36, … The first common multiple is 36, so the LCM of 9 and 12 is 36.

Multiples of 6 and of 8 first meet at 24.Multiples of 6Multiples of 8612182481624First meeting point: 24, so LCM of 6 and 8 = 24

In the figure : In the figure multiples of 6 and 8 first meet at 24.

  • Easy : Easy: LCM of 3 and 4: multiples 3, 6, 9, 12 and 4, 8, 12; LCM = 12.
  • Medium : Medium: LCM of 6 and 10: 6, 12, 18, 24, 30 and 10, 20, 30; LCM = 30.
  • Exam-type : Exam: LCM of 15 and 20: multiples of 20 are 20, 40, 60 and 60 ÷ 15 = 4, so LCM = 60.

Rule 2: Prime factor method and meeting-again problems — For each prime, take the greatest number of times it appears in any one number, then multiply. 12 = 2×2×3 and 18 = 2×3×3 give 2×2×3×3 = 36. Use the LCM for "when will they meet again".

  • Easy : Easy: 6 = 2×3 and 15 = 3×5 give 2×3×5 = 30.
  • Medium : Medium: 20 = 2×2×5 and 30 = 2×3×5 give 2×2×3×5 = 60.
  • Exam-type : Exam: Buses leave together at 6:00 a.m. and then every 12, 18 and 30 minutes. LCM = 2×2×3×3×5 = 180 minutes, so they meet again at 9:00 a.m.
Remember

LCM = smallest common multiple. Highest count of each prime. "Meet again" = LCM.

Common mistake

Using the GCD for "meet again" problems. Use the LCM there.

Shortcut

Memory clue: "मसावि = कापणे (लहान तुकडे); लसावि = भेटणे (पुन्हा एकत्र)" (GCD = cutting into pieces, LCM = meeting again).

Spot the difference

QuestionUse
Cut into longest equal piecesGCD
When will they meet againLCM

Practice questions

Easy
  1. 1Find the LCM of 4 and 6.
  2. 2Find the LCM of 5 and 8.
  3. 3Find the LCM of 9 and 15.
Medium
  1. 4Find the LCM of 12 and 16.
  2. 5Find the LCM of 6, 8 and 12.
  3. 6Bells ring every 15 and 25 minutes. After how many minutes do they ring together?
Challenging
  1. 7Two lights blink every 8 and 10 seconds and blink together now. When do they blink together next?
  2. 8Why is the LCM of 7 and 11 equal to 77?
Show answers (try them yourself first)
  1. 112.
  2. 240.
  3. 345.
  4. 448.
  5. 56 = 2×3, 8 = 2×2×2, 12 = 2×2×3, so LCM = 2×2×2×3 = 24.
  6. 6LCM of 15 and 25 is 75 minutes.
  7. 7After the LCM of 8 and 10, which is 40 seconds.
  8. 8They are different primes with no common factor, so the LCM is their product.

GCD and LCM together

From daily life

A student found the GCD and LCM of 12 and 20 and noticed something nice: 12 × 20 = 4 × 60. The GCD and the LCM are tied together by one rule.

Simple definition

For two numbers: number 1 × number 2 = GCD × LCM.

Rule 1: The rule — For 12 and 20, the GCD is 4 and the LCM is 60. Product of the numbers = 12 × 20 = 240 and GCD × LCM = 4 × 60 = 240.

For 18 and 30: the product 540 equals GCD 6 times LCM 90.The two numbers18 × 30 = 540GCD × LCM6 × 90 = 540=Product of two numbers = GCD × LCMSo LCM = 360 ÷ 6 = 60 when product 360, GCD 6

In the figure : In the figure the product of two numbers = GCD × LCM.

  • Easy : Easy: 8 and 12: GCD 4, LCM 24; 8 × 12 = 96 = 4 × 24.
  • Medium : Medium: 15 and 25: GCD 5, LCM 75; 15 × 25 = 375 = 5 × 75.
  • Exam-type : Exam: Two numbers 36 and 60 have GCD 12. Then LCM = (36 × 60) ÷ 12 = 2160 ÷ 12 = 180.

Rule 2: Finding a missing value — If three of the four values are known, find the fourth. Numbers 16 and ?, GCD 4, LCM 80: the other number = (4 × 80) ÷ 16 = 20. Facts: the GCD divides the LCM, and the LCM is never smaller than the larger number.

  • Easy : Easy: Numbers 10 and 15, GCD 5: LCM = 150 ÷ 5 = 30.
  • Medium : Medium: GCD 4, LCM 80, one number 16: the other = 320 ÷ 16 = 20.
  • Exam-type : Exam: Co-prime 7 and 9: GCD 1, LCM = 7 × 9 ÷ 1 = 63.
Remember

Product = GCD × LCM (for two numbers only). GCD ≤ smaller number ≤ larger number ≤ LCM.

Common mistake

Using this rule for three numbers. It works for two numbers only.

Shortcut

Memory clue: "दोन संख्यांचा गुणाकार = मसावि × लसावि".

Spot the difference

ItemRule
Producta × b = GCD × LCM
Co-primeLCM = a × b
GCD and LCMGCD divides LCM

Practice questions

Easy
  1. 1For 6 and 8, GCD is 2 and LCM is 24. Verify the rule.
  2. 2Numbers 9 and 12 have GCD 3. Find the LCM.
  3. 3GCD 5, LCM 60 and one number 15. Find the other number.
Medium
  1. 4Find the LCM of the co-prime numbers 5 and 8.
  2. 5Can the LCM of two numbers be smaller than the larger number?
  3. 6GCD 6, LCM 36, one number 12. Find the other number and check.
Challenging
  1. 7For 14 and 21, find GCD, LCM and verify the rule.
  2. 8Can the GCD of two numbers be larger than their LCM? Why?
Show answers (try them yourself first)
  1. 16 × 8 = 48 and 2 × 24 = 48.
  2. 2108 ÷ 3 = 36.
  3. 3(5 × 60) ÷ 15 = 20.
  4. 440.
  5. 5No, the LCM is always at least the larger number.
  6. 6(6 × 36) ÷ 12 = 18. GCD of 12 and 18 is 6 and LCM is 36 ✓.
  7. 7GCD 7, LCM 42; 14 × 21 = 294 and 7 × 42 = 294.
  8. 8No. The GCD divides the LCM, so it cannot be larger.