The HCF of two numbers is 20 and their sum is 100 the possible numbers are

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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