LCM by Prime Factorization

LCM by Prime Factorization finds the least common multiple of two or more numbers by expressing each number as a product of primes. Select every prime factor appearing in the numbers, using its greatest power, and multiply these selected factors. This method gives a systematic way to solve LCM prime factorization questions.

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What Is LCM by Prime Factorization?

LCM by Prime Factorization is a method of finding the least common multiple by writing each number as a product of prime factors. The LCM is the product of every distinct prime factor raised to the highest power found in any of the numbers.

For example, 12 = 2² × 3 and 18 = 2 × 3². The prime factors are 2 and 3. Their highest powers are 2² and 3², so LCM = 2² × 3² = 36. The result is the smallest positive number divisible by both 12 and 18.

LCM by Prime Factorization Formula & Tricks

Important Formulas

Highest-power rule
LCM = product of each distinct prime factor with its greatest exponent

Write the prime factorization of every number, compare the exponents of each prime, and use the highest exponent.

Two-number product relation
LCM(a, b) × HCF(a, b) = a × b

For two positive integers, the product of their LCM and HCF equals the product of the numbers.

Coprime numbers
If HCF(a, b) = 1, then LCM(a, b) = a × b

Coprime numbers have no common prime factor, so all their prime factors are included in the LCM.

Quick Tricks

Use the greatest exponent only

When the same prime appears in more than one factorization, retain its largest power and do not multiply all repeated powers.

Example: For 2³ × 3 and 2 × 3², use 2³ and 3². Thus, LCM = 2³ × 3² = 72.
Check divisibility by every given number

After calculating the LCM, divide it by each original number. A remainder of zero confirms that the result is a common multiple.

Example: For 12 and 18, LCM = 36. Since 36 ÷ 12 = 3 and 36 ÷ 18 = 2, 36 is a common multiple.
Remove numbers already represented by others

If one number divides another, the larger number already contains all prime factors and powers of the smaller number. The larger number alone determines the LCM for that pair.

Example: Since 8 divides 24, LCM(8, 24) = 24.

LCM by Prime Factorization Concepts

Prime Factorization Method

To find the LCM, first express every given number as a product of prime factors, then choose the highest power of each distinct prime.

Arrange the factorizations in rows or columns. Compare the powers of 2, 3, 5 and other primes. Multiply one selected power for each distinct prime. For 20 = 2² × 5 and 28 = 2² × 7, select 2², 5 and 7. Therefore, LCM = 2² × 5 × 7 = 140.

Example: LCM(20, 28) = 2² × 5 × 7 = 140.

Repeated Prime Factors and Highest Powers

A prime factor is included with its greatest exponent among all the given numbers.

If a prime occurs as 2, 2² and 2⁴ in different factorizations, use 2⁴ in the LCM. Lower powers are already factors of the greatest power and need not be multiplied separately. For 24 = 2³ × 3 and 40 = 2³ × 5, the selected powers are 2³, 3 and 5.

Example: LCM(24, 40) = 2³ × 3 × 5 = 120.

LCM of Three or More Numbers

For three or more numbers, include the highest power of every prime appearing in any factorization.

Factor each number separately and compare all prime powers. For 18 = 2 × 3², 24 = 2³ × 3 and 30 = 2 × 3 × 5, choose 2³, 3² and 5. Their product gives the LCM.

Example: LCM(18, 24, 30) = 2³ × 3² × 5 = 360.

Coprime Numbers and Common Factors

If two numbers have HCF 1, their LCM equals their product; if they share factors, common factors are included only once at their highest power.

For 8 and 15, the prime factorizations are 2³ and 3 × 5, with no common prime factor. Hence, LCM = 2³ × 3 × 5 = 120. For numbers with common factors, the highest-power rule prevents repeated inclusion of the same prime.

Example: LCM(8, 15) = 8 × 15 = 120 because 8 and 15 are coprime.

LCM by Prime Factorization Video Lessons

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LCM Using Prime Factorisation and Ladder Method

Learn to find the least common multiple using prime factorisation and the ladder method, with clear steps for applying both techniques to number system problems.

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Practice LCM by Prime Factorization Questions

Practise published questions related to this topic.

1√(0.0225/0.04) = ?→ 2What is the least number which should be added to 539, so that the number obtained is exactly divisible by 10, 15 and 3?→ 3A pharmaceutical quality analyst is verifying the accuracy of a compound used in tablet production. To ensure uniformity in dosage, a specific concentration index is calculated using the formula: √()((9.5 × .085) /(.0017 × .19)) What is the resulting value of this expression, which determines whether the batch meets the required standards?→ 4What is the smallest number that must be added to 45873 so that the sum is divisible by 11?→ 5Simplify: √(2061 - √(1268 + √(850 - 66)))→ 6Which of the following number is divisible by both 9 and 11?→ 7What is the L.C.M. of 45, 63 and 81?→ 8What is the Highest Common Factor of 0.48, 0.64 and 0.80?→ 9The product of Highest Common Factor and Least Common Multiple of 2 numbers is 6930. One of the numbers is 154. What is the other number?→ 10Least Common Multiple of A and B is 36 and C = 24. What is the Least Common Multiple of A, B and C?→

LCM by Prime Factorization Quick Quiz

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Quick Revision Notes

LCM by Prime Factorization: Quick Revision

Use these rules to calculate an LCM from prime factors.

  • Prime-factorize every given number.
  • List each distinct prime appearing in any factorization.
  • Select the greatest exponent of each prime.
  • Multiply the selected prime powers to obtain the LCM.
  • A number that divides another does not add a new factor beyond the larger number's factorization.
  • For two numbers, LCM × HCF = product of the two numbers.
  • For coprime numbers, LCM equals the product of the numbers.
  • Verify the answer by checking that every given number divides the calculated LCM.

LCM by Prime Factorization FAQs

What is the formula for LCM by prime factorization?

LCM = product of each distinct prime factor raised to its greatest exponent in any of the given numbers.

How do you find the LCM of 12 and 30 using prime factors?

12 = 2² × 3 and 30 = 2 × 3 × 5. Select 2², 3 and 5, so LCM = 2² × 3 × 5 = 60.

Why is the highest power used in the LCM?

The highest power contains every lower power of the same prime as a factor. Therefore, selecting the highest power makes the number divisible by all the given numbers without unnecessary repetition.

What is the LCM of 2³ × 3 and 2 × 3²?

Use the greatest powers 2³ and 3². Therefore, LCM = 2³ × 3² = 72.

How is LCM found when one number divides another?

The larger number is the LCM. For example, since 12 divides 36, LCM(12, 36) = 36.

What is the LCM of two coprime numbers?

It is their product because coprime numbers have no common prime factor. For example, LCM(9, 20) = 9 × 20 = 180.

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