LCM Concepts

1. LCM of Numbers

• Idea: Finding the smallest positive number that is exactly divisible by two or more given numbers.

• Formula:

$$LCM(a,b)=\frac{a\times b}{HCF(a,b)}$$

• Example: Find the LCM of 15, 20 and 30.

  Solution:

  15 = 3 × 5
  20 = 2² × 5
  30 = 2 × 3 × 5

  Taking the highest power of each prime:

$$LCM=2^2\times3\times5=60$$

Answer: 60

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2. Smallest Number Divisible by Given Numbers

• Idea: Finding the smallest number that is divisible by several given divisors.

• Formula:

$$\text{Smallest number}=LCM(\text{given divisors})$$

• Example: Find the smallest number divisible by 12, 18 and 30.

  Solution:

$$LCM(12,18,30)=180$$

Answer: 180

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3. Smallest Number Leaving the Same Remainder

• Idea: Finding the smallest number that leaves the same remainder when divided by several numbers.

• Formula:

$$N=LCM(a,b,c)+r$$

where (r) is the common remainder.

• Example: Find the smallest number which leaves remainder 3 when divided by 8, 12 and 20.

  Solution:

$$LCM(8,12,20)=120$$

Therefore,

$$N=120+3=123$$

Answer: 123

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4. Smallest Number Leaving Different Remainders

• Idea: Finding the smallest number when different divisors produce specified remainders.

• Formula:

  Convert the conditions into congruences:

$$N\equiv r_1\pmod {a}$$ $$N\equiv r_2\pmod {b}$$

and solve for the smallest (N).

• Example: Find the smallest number which leaves remainder 2 when divided by 5 and remainder 3 when divided by 7.

  Solution:

  Numbers leaving remainder 2 when divided by 5:

  2, 7, 12, 17, 22, ...

  Check division by 7:

  17 ÷ 7 → remainder 3

  Answer: 17

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5. Cyclic Events / Repeated Events

• Idea: Finding when events that repeat at fixed intervals will occur together again.

• Formula:

$$\text{Next common occurrence}=LCM(\text{intervals})$$

• Example: Two traffic lights change every 24 seconds and 36 seconds. When will they change together again?

  Solution:

$$LCM(24,36)=72$$

Answer: 72 seconds

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6. Bells, Alarms & Timetable Problems

• Idea: Finding when bells, alarms, machines or buses operating at different intervals coincide again.

• Formula:

$$\text{Common interval}=LCM(t_1,t_2,t_3,\ldots)$$

• Example: Three bells ring every 12, 18 and 30 minutes. If they ring together at 9:00 AM, when will they next ring together?

  Solution:

$$LCM(12,18,30)=180\text{ minutes}$$ $$180\text{ minutes}=3\text{ hours}$$

Answer: 12:00 PM

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7. LCM of Fractions

• Idea: Finding the LCM of fractions.

• Formula:

$$LCM\left(\frac{a}{b},\frac{c}{d}\right) ======================================= \frac{LCM(a,c)}{HCF(b,d)}$$

• Example: Find the LCM of

$$\frac34,\frac56,\frac78$$

Solution:

$$LCM(3,5,7)=105$$ $$HCF(4,6,8)=2$$

Therefore,

$$LCM=\frac{105}{2}$$

Answer: (\frac{105}{2})

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8. LCM of Decimals

• Idea: Converting decimals into integers before finding the LCM.

• Formula:

  Multiply all numbers by the same power of 10, find the LCM, then divide by that power of 10.

• Example: Find the LCM of 0.6, 0.8 and 1.2.

  Solution:

  Multiply by 10:

  0.6 → 6
  0.8 → 8
  1.2 → 12

$$LCM(6,8,12)=24$$

Divide by 10:

$$LCM=2.4$$

Answer: 2.4

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Advanced

9. LCM with a Common Difference

• Idea: Finding the smallest number that satisfies several divisibility conditions involving numbers in a sequence.

• Formula:

  Use the LCM of the relevant divisors and then apply the required remainder or sequence condition.

• Example: Find the smallest number divisible by 6, 8 and 15 and greater than 500.

  Solution:

$$LCM(6,8,15)=120$$

Multiples of 120 greater than 500:

600, 720, 840, ...

Answer: 600

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10. LCM–HCF Relationship

• Idea: Finding an unknown number when the HCF, LCM and one number are given.

• Formula:

$$HCF(a,b)\times LCM(a,b)=a\times b$$

• Example: The HCF of two numbers is 12 and their LCM is 180. If one number is 36, find the other.

  Solution:

$$12\times180=36\times x$$ $$x=60$$

Answer: 60