# Example 1

Find the HCF of 108, 144, 180.

Find the factors of the given numbers.

108 = 2Â² x 3Â³

144 = 24 x 3Â²

180 = 2Â² x 3Â² x 5

HCF = (Choose the lowest powers to the prime factors) = 2 x 2 x 3 x 3 = 36.

# Example 2

Find the LCM of 12, 18, 30.

List down the factors of the given numbers as follows.

12 = 2Â² x 3

18 = 2 x 3Â²

30 = 2 x 3 x 5

So LCM = (Choose the highest powers of the prime factors) = 2Â² x 3Â² x 5 = 180.

# Example 3

Find the least number which when divided by 4, 8, 12, 16, 20, 48 and 80 will leave in each case remainder 3.

In this case the numbers 4, 8, 12 and 16 are factors of 48, so these can be left.

Similarly 20 can be left, being a factor of 80.

The remaining numbers are 48 and 80 and LCM of these two numbers would be the LCM of all these numbers.

LCM of 48 and 80 is 240 (by the above method).

Because a remainder of 3 is required so it will be added in the LCM i.e. 240 + 3 = 243 is the answer.

# Example 4

What is the smallest number which when divided by 2, 3, 4, 5 and 6 respectively leaves a remainder of 1, 2, 3, 4 and 5 respectively?

It can be seen that in this case the difference between the number and its respective remainder is all equal i.e. 2 â€“ 1 = 1, 3 â€“ 2 = 1, 4 â€“ 3 = 1,

and so on.

First the LCM of 2, 3, 4, 5 and 6 would be calculated out of which we can leave 2 and 3, being the factors of other numbers.

LCM of 4, 5 and 6 is 60.

From this the common difference which is equal to one is subtracted from it.

Hence 60 â€“ 1 = 59 is the answer.

To check, se see that 59 when divided by any of the given numbers, yields a remainder 1.

# Example 5

Find the smallest number which when divided by 15 leaves remainder 5, when divided by 25 leaves remainder 15, when divided by 35

leaves remainder 25.

Notice that the difference between the number and respective remainder is equal.

15 â€“ 5 = 10, 25 â€“ 15 = 10, 35 â€“ 25 = 10.

First, the LCM of 15, 25 and 35 will be calculated which is 525.

Then we subtract from it the common difference i.e. 10. Hence 525 â€“ 10 = 515 is the answer.

# Example 6

The LCM of two numbers is 4800 and their GCM is 160. If one number is 480 then the second number is?

(Product of two numbers

= Product of their HCF and LCM)

= 4800 Ã— 160 = 480 Ã— N.

Hence N = 1600.

# Example 7

Find the LCM and HCF of 2/3 and 6/7

LCM of the fractions = LCM of Numerators = 2 & 6 = 6

HCF of Denominators = 3 & 7 = 1

So LCM = 6/1 = 6

HCF of the fractions = HCF of Numerators = 2 & 6 = 2

LCM of Denominators = 3 & 7 = 21

So HCF = 2/21

# Example 8

Find the HCF and of 1.75, 5.6 and 7.

Without decimal point, these numbers are 175, 560 and 700. HCF of 175, 560 and 700 is 35.

Hence HCF of 1.75, 5.6 and 7 is 0.35.

# Example 9

Find the LCM of 5x^{2}y^{3}z^{5} and 3xy^{2}z^{7}.

LCM is the product of all prime factors of the given numbers, the common factors among them being in their highest degree. e.g, will be

5*3*x^{2}y^{3}z^{7} = 15x^{2}y^{3}z^{7}, where x, y and z are the prime factors.

# Example 10

Three drums contain 36 litres, 45 litres and 72 litres of oil. What biggest measure can measure all the different quantities exactly?

Biggest Measure = (HCF of 36, 45, 72) litres = 9 litres