## Topic outline

• ### Playing With Numbers (Some Problems on HCF and LCM)

• ### Some Problems on HCF and LCM

#### Example 1: Two milk tankers contain 450 litres and 600 litres of milk respectively. Find the maximum capacity of a container which can measure the milk of both the tankers when used an exact number of times.

Solution: To find maximum capacity of a container, we will find the HCF of 450 and 600.

 2 450, 600 3 225, 300 5 75,  100 5 15,   20 3,     4

HCF = 2 x 3 x 5 x 5 = 150

Therefore, maximum capacity of the required container is 150 litres. It will fill the first container in 450 ÷ 150 = 3 and the second in 600 ÷ 150 = 4 refills.

#### Example 2: Find the least number which when divided by 12, 15, 18 and 20 leaves a remainder 5 in each case.

Solution: Here, we will find LCM of 12, 15, 18 and 20.

 2 12, 15, 18, 20 2 6,  15,  9,  10 3 3,  15,  9,   5 3 1,   5,   3,   5 5 1,   5,   1,   5 1,   1,   1,   1

Thus, LCM = 2 x 2 x 3 x 3 x 5 = 180

180 is the least number which when divided by the given numbers will leave remainder 0 in each case.

Therefore, the required number is 5 more than 180. The required least number = 180 + 5 = 185

#### Example 3: Find the smallest 4 digit number which is divisible by 18, 24 and 32.

Solution: Here, we will find LCM of 18, 24 and 32.

 2 18, 24, 32 2 9,  12, 16 2 9,   6,   8 2 9,   3,   4 2 9,   3,   2 3 9,   3,   1 3 3,   1,   1 1,   1,  1

Thus, LCM = 2 x 2 x 2 x 2 x 2 x 3 x 3 = 288

We need to find smallest 4 digit number multiple of 288

288 x 1 = 288

288 x 2 = 576

288 x 3 = 864

288 x 4 = 1152

288 x 5 = 1440

Therefore, the smallest 4 digit divisible by 18, 24 and 32 is 1152

#### Example 4: Determine the greatest 3-digit number exactly divisible by 7, 9 and 12.

Solution: Here, we will find LCM of 7, 9 and 12.

 2 7, 9, 12 2 7, 9,  6 3 7, 9,  3 3 7, 3,  1 7 7, 1,  1 1, 1,  1

Thus, LCM = 2 x 2 x 3 x 3 x 7 = 252

We need to find greatest 3 digit number multiple of 252

252 x 1 = 252

252 x 2 = 504

252 x 3 = 756

252 x 4 = 1008

Therefore, the greatest 3 digit divisible by 7, 9 and 12 is 756.

#### Example 5: The length, breadth and height of a box are 75cm, 85cm and 95cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.

Solution: Longest tape = HCF of 75cm, 85cm and 95cm.

 5 75, 85, 95 15, 17, 19

Therefore, longest tape = 5cm

#### Example 6: In a morning walk, three girls step off together from the same spot. Their steps measure 66cm, 78cm and 90cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps?

Solution: The distance covered by each one of them is required to be the same as well as minimum. The required minimum distance each should walk would be the lowest common multiple of the measures of their steps. Thus, we find the LCM of 66, 78 and 90.

 2 66, 78, 90 2 33, 36, 45 2 33, 18, 45 3 33,  9,  45 3 11,  3,  15 5 11,  1,   5 11 11,  1,   1 1,   1,   1

LCM = 2 x 2 x 2 x 3 x 3 x 5 x 11 = 3960cm.

The required minimum distance is 3960cm.

#### Example 7: HCF and LCM of two numbers is 10 and 60 respectively. One of the numbers is 20, find the other number.

Solution: H.C.F of two numbers = 10

L.C.M of two numbers = 60

One of the number = 20

Other number = H.C.F x L.C.M / One of the number

= 10x60 / 20

= 600/20 30

#### Example 8: HCF of two numbers is 2 and product of two numbers is 2340. Find the LCM of two numbers.

Solution: H.C.F of two numbers = 2

Product of the two numbers = 2340

L.C.M = Product of two numbers/ H.C.F

= 2340/2 1170

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