HCF of two numbers is 15 and their LCM is 105 if one of the numbers is 5 then other number is

LCM of 5 and 15 is the smallest number among all common multiples of 5 and 15. The first few multiples of 5 and 15 are (5, 10, 15, 20, 25, 30, . . . ) and (15, 30, 45, 60, 75, 90, 105, . . . ) respectively. There are 3 commonly used methods to find LCM of 5 and 15 - by listing multiples, by division method, and by prime factorization.

Table of Contents Show

What is the LCM of 5 and 15?
Methods to Find LCM of 5 and 15
LCM of 5 and 15 by Prime Factorization
LCM of 5 and 15 by Division Method
LCM of 5 and 15 by Listing Multiples
Which of the following is the LCM of 5 and 15? 3, 28, 15, 12
What is the Least Perfect Square Divisible by 5 and 15?
If the LCM of 15 and 5 is 15, Find its GCF.
What is the Relation Between GCF and LCM of 5, 15?

What is the LCM of 5 and 15?

Answer: LCM of 5 and 15 is 15.

Explanation:

The LCM of two non-zero integers, x(5) and y(15), is the smallest positive integer m(15) that is divisible by both x(5) and y(15) without any remainder.

Methods to Find LCM of 5 and 15

Let's look at the different methods for finding the LCM of 5 and 15.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 5 and 15 by Prime Factorization

Prime factorization of 5 and 15 is (5) = 51 and (3 × 5) = 31 × 51 respectively. LCM of 5 and 15 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 31 × 51 = 15.
Hence, the LCM of 5 and 15 by prime factorization is 15.

LCM of 5 and 15 by Division Method

To calculate the LCM of 5 and 15 by the division method, we will divide the numbers(5, 15) by their prime factors (preferably common). The product of these divisors gives the LCM of 5 and 15.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 5 and 15. Write this prime number(3) on the left of the given numbers(5 and 15), separated as per the ladder arrangement.
Step 2: If any of the given numbers (5, 15) is a multiple of 3, divide it by 3 and write the quotient below it. Bring down any number that is not divisible by the prime number.
Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 5 and 15 is the product of all prime numbers on the left, i.e. LCM(5, 15) by division method = 3 × 5 = 15.

LCM of 5 and 15 by Listing Multiples

To calculate the LCM of 5 and 15 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 5 (5, 10, 15, 20, 25, 30, . . . ) and 15 (15, 30, 45, 60, 75, 90, 105, . . . . )
Step 2: The common multiples from the multiples of 5 and 15 are 15, 30, . . .
Step 3: The smallest common multiple of 5 and 15 is 15.

∴ The least common multiple of 5 and 15 = 15.

☛ Also Check:

Example 1: Find the smallest number that is divisible by 5 and 15 exactly.

Solution:

The smallest number that is divisible by 5 and 15 exactly is their LCM.
⇒ Multiples of 5 and 15:
- Multiples of 5 = 5, 10, 15, 20, 25, 30, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, 90, . . . .
Therefore, the LCM of 5 and 15 is 15.
Example 2: The GCD and LCM of two numbers are 5 and 15 respectively. If one number is 15, find the other number.

Solution:

Let the other number be a.
∵ GCD × LCM = 15 × a ⇒ a = (GCD × LCM)/15 ⇒ a = (5 × 15)/15 ⇒ a = 5
Therefore, the other number is 5.

Example 3: The product of two numbers is 75. If their GCD is 5, what is their LCM?

Solution:

Given: GCD = 5 product of numbers = 75 ∵ LCM × GCD = product of numbers ⇒ LCM = Product/GCD = 75/5 Therefore, the LCM is 15.

The probable combination for the given case is LCM(5, 15) = 15.

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The LCM of 5 and 15 is 15. To find the least common multiple of 5 and 15, we need to find the multiples of 5 and 15 (multiples of 5 = 5, 10, 15, 20; multiples of 15 = 15, 30, 45, 60) and choose the smallest multiple that is exactly divisible by 5 and 15, i.e., 15.

Which of the following is the LCM of 5 and 15? 3, 28, 15, 12

The value of LCM of 5, 15 is the smallest common multiple of 5 and 15. The number satisfying the given condition is 15.

What is the Least Perfect Square Divisible by 5 and 15?

The least number divisible by 5 and 15 = LCM(5, 15) LCM of 5 and 15 = 3 × 5 [Incomplete pair(s): 3, 5]

⇒ Least perfect square divisible by each 5 and 15 = LCM(5, 15) × 3 × 5 = 225 [Square root of 225 = √225 = ±15]

Therefore, 225 is the required number.

If the LCM of 15 and 5 is 15, Find its GCF.

LCM(15, 5) × GCF(15, 5) = 15 × 5 Since the LCM of 15 and 5 = 15 ⇒ 15 × GCF(15, 5) = 75

Therefore, the GCF (greatest common factor) = 75/15 = 5.

What is the Relation Between GCF and LCM of 5, 15?

The following equation can be used to express the relation between GCF and LCM of 5 and 15, i.e. GCF × LCM = 5 × 15.

Given:

L.C.M = 105 and H.C.F = 7

First number = 21

Formula used:

L.C.M × H.C.F = Product of numbers

Calculation:

Let the second number be 'n'

⇒ 105 × 7 = 21 × n

⇒ n = 735/21 = 35

∴ The second number is 35.

Important Points

In arithmetic and number theory, the least common multiple, the lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b.

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We will learn the relationship between H.C.F. and L.C.M. of two numbers.

First we need to find the highest common factor (H.C.F.) of 15 and 18 which is 3.

Then we need to find the lowest common multiple (L.C.M.) of 15 and 18 which is 90.

H.C.F. × L.C.M. = 3 × 90 = 270

Also the product of numbers = 15 × 18 = 270

Therefore, product of H.C.F. and L.C.M. of 15 and 18 = product of 15 and 18.

Again, let us consider the two numbers 16 and 24

Prime factors of 16 and 24 are:

16 = 2 × 2 × 2 × 2

24 = 2 × 2 × 2 × 3

L.C.M. of 16 and 24 is 48;

H.C.F. of 16 and 24 is 8;

L.C.M. × H.C.F. = 48 × 8 = 384

Product of numbers = 16 × 24 = 384

So, from the above explanations we conclude that the product of highest common factor (H.C.F.) and lowest common multiple (L.C.M.) of two numbers is equal to the product of two numbers

or, H.C.F. × L.C.M. = First number × Second number

or, L.C.M. = \(\frac{\textrm{First Number} \times \textrm{Second Number}}{\textrm{H.C.F.}}\)

or, L.C.M. × H.C.F. = Product of two given numbers

or, L.C.M. = \(\frac{\textrm{Product of Two Given Numbers}}{\textrm{H.C.F.}}\)

or, H.C.F. = \(\frac{\textrm{Product of Two Given Numbers}}{\textrm{L.C.M.}}\)

Solved examples on the relationship between H.C.F. and L.C.M.:

1. Find the L.C.M. of 1683 and 1584.

Solution:

First we find highest common factor of 1683 and 1584

Therefore, highest common factor of 1683 and 1584 = 99

Lowest common multiple of 1683 and 1584 = First number × Second number/ H.C.F.

= \(\frac{1584 × 1683}{99}\)

= 26928

2. Highest common factor and lowest common multiple of two numbers are 18 and 1782 respectively. One number is 162, find the other.

Solution:

We know, H.C.F. × L.C.M. = First number × Second number then we get,

18 × 1782 = 162 × Second number

\(\frac{18 × 1782}{162}\) = Second number

Therefore, the second number = 198

3. The HCF of two numbers is 3 and their LCM is 54. If one of the numbers is 27, find the other number.

Solution:

HCF × LCM = Product of two numbers

3 × 54 = 27 × second number

Second number = \(\frac{3 × 54}{27}\)

Second number = 6

4. The highest common factor and the lowest common multiple of two numbers are 825 and 25 respectively. If one of the two numbers is 275, find the other number.

Solution:

We know, H.C.F. × L.C.M. = First number × Second number then we get,

825 × 25 = 275 × Second number

\(\frac{825 × 25}{275}\) = Second number

Therefore, the second number = 75

5. Find the H.C.F. and L.C.M. of 36 and 48.

Solution:

H.C.F. = 2 × 2 × 3 = 12

L.C.M. = 2 × 2 × 3 × 3 × 4 = 144

H.C.F. × L.C.M. = 12 × 144 = 1728

Product of the numbers = 36 × 48 = 1728

Therefore, product of the two numbers = H.C.F × L.C.M.

2. The H.C.F. of two numbers 30 and 42 is 6. Find the L.C.M.

Solution:

We have H.C.F. × L.C.M. = product of the numbers

6 × L.C.M. = 30 × 42

L.C.M. = \[\frac{30 × 42}{\sqrt{6}}\]

= \[\frac{1260}{\sqrt{6}}\]

= 210

3. Find the greatest number which divides 105 and 180 completely.

Solution:

The greatest number here is the H.C.F of 105 and 180

Common factors are 5, 3

H.C.F. = 5 × 3 = 15

Therefore, the greatest number that divides 105 and 180 completely is 15.

4. Find the least number which leaves 3 as remainder when divided by 24 and 42.

Solution:

L.C.M. of 24 and 42 leaves no remainder when divided by the number 24 and 42.

L.C.M. = 2 × 3 × 4 × 7 = 168

The least number which leaves 3 as remainder is 168 + 3 = 171.

Important Notes:

Two numbers which have only 1 as the common factor are called co-prime.

The least common multiple (L.C.M.) of two or more numbers is the smallest number which is divisible by all the numbers.

If two numbers are co-prime, their L.C.M. is the product of the numbers.

If one number is the multiple of the other, then the multiple is their L.C.M.

L.C.M. of two or more numbers cannot be less than any one of the given numbers.

H.C.F. of two or more numbers is the highest number that can divide the numbers without leaving any remainder.

If one number is a factor of the second number then the smaller number is the H.C.F. of the two given numbers.

The product of L.C.M. and H.C.F. of two numbers is equal to the product of the two given numbers.

Questions and Answers on Relationship between H.C.F. and L.C.M.

1. The H.C.F. of two numbers 20 and 75 is 5. Find their L.C.M.

2. The L.C.M. of two numbers 72 and 180 is 360. Find their H.C.F.

3. The L.C.M. of two numbers is 120. If the product of the numbers is 1440, find the H.C.F.

4. Find the least number which leaves 5 as remainder when divided by 36 and 54.

5. The product of two numbers is 384. If their H.C.F. is 8, find the L.C.M.

Answer:

1. 300

2. 36

3. 12

4. 113

5. 48

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