Word Problems Involving Highest Common Factor and Least Common Multiple has several questions on finding LCM of Two or More Numbers, HCF of Two or More Numbers, Relationship between H.C.F. and L.C.M, and many more. Before attempting the actual exam students are advised to solve these kinds of Highest Common Factor and Least Common Multiple Worksheets to be familiar with various models being asked in exams. Try to solve the HCF and LCM Questions on your own and then verify with our solutions explained clearly.

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**Example 1.
**The length, breadth, height of a room are 4 m 50 cm, 6 m 30 cm, and 8 m 20 cm respectively. Find the longest tape which can measure the dimensions of the room exactly?

**Solution:**

First, we have to convert the dimensions into cm.

We know that 1 m=100 cm.

Given,

The length of the room=4 m 50 cm=4 (100) cm +50 cm=450 cm

The breadth of the room=6 m 30 cm=6(100)cm+30 cm=630 cm

The height of the room=8 m 20 cm=8(100)cm+20 cm=820 cm

HCF(450,630,820)=longest tape that can measure the dimensions of the room.

HCF(450,630,820)=10 cm

**Example 2.
**Find the least length of a rope which can be cut into the whole number of pieces of lengths 34 cm, 52 cm, and 78 cm?

**Solution:**

Given,

length of the pieces=34 cm,52 cm,78 cm

LCM(34,52,78)

34=2 × 17

52=2 × 2 × 13

78=2 × 3 × 13

=2^{2} × 17 × 13 × 3

=2652 cm

Therefore, the length of the rope is 2652 cm.

**Example 3.
**The product of the two numbers is 5600. If their HCF is 10. Find their LCM?

**Solution:**

Given,

The product of the two numbers is= 5600

HCF=10

We know that product of two numbers=LCM × HCF

5600=LCM × 10

LCM=5600/10

=560

Therefore, LCM is 560.

**Example 4.
**Find the multiple of 80 which lies between 400 to 600 which has a tens digit is double than one’s digit?

**Solution:**

Multiples of 80 are 80,160,240,320,400,480,560,640 etc.

Given,

The number is between 400 to 600 and the tens digit is double than the hundreds digit.

In 480, the tens digit is double than the hundreds digit.

Hence, 480 is the number.

**Example 5.
**If the HCF and LCM of two numbers are 10, 840. If one of their numbers is 70. Find the other number?

**Solution:**

Given,

HCF=10

LCM of two numbers=840

One of their numbers=70

We know that product of two numbers=LCM × HCF

70. x=840 × 10

x=8400/70=120

Therefore, the other number is 120.

**Example 6.
**Find the largest number that divides 184, 244 leaving 4 as the remainder?

**Solution:**

Given,

The number divides 184 and leaves 4 as the remainder

Therefore, the number divides 184 – 4 = 180 exactly

The number also divides 244 leaving 4 as the remainder

Therefore, the number divides 244 – 4 = 240 exactly

Now we have to find the highest common factor H.C.F of 180 and 240

Hence, the largest number is 60.

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**Example 7.
**90 boys, 70 girls, 10 teachers are equally arranged in buses such that no one is left. Find the highest number of buses required for them?

**Solution:**

Given,**
**No. of boys going for the trip=90

No. of girls going for the trip=70

No. of teachers going for the trip=10

The highest number of buses required=HCF(90,70,10)

Factors of 90 are 1,2,3,5,6,9,10,15,18,30,45,90

Factors of 70 are 1,2,5,7,10,14,35,70

Factors of 10 are 1,2,5,10

GCF=10

Therefore, 10 buses required for the trip.

**Example 8.
**The chairs in the auditorium can be arranged in rows of 43,65. Find the least number of chairs in the auditorium?

**Solution:**

Given,

The chairs can be arranged in rows=43, 65

The least number of chairs in the auditorium=LCM(43,65)

43=43

65=5 × 13

LCM(43,65)= 43 × 5 × 13

=2,795

**Example 9.
**Janu wants to plant 32 red rose plants and 48 white rose plants. What is the greatest number of rows possible if each row has the same number of red rose plants and white rose plants?

**Solution:**

Given,**
**Janu wants to plant red rose plants=32

Janu want to plant white rose plants=48

We have to find the greatest number of rows and each row should have the same number of red rose plants and white rose plants.

=HCF(32,48)

Factors of 32 are 1,2,4,8,16,32.

Factors of 48 are 1,2,3,4,6,8,12,16,24,48

HCF(32,48)=16

Hence, there will be 16 rows.

**Example 10.
**Find the multiple of 90 which lies between 200 and 500 where the digits at tens place and hundreds place are consecutive numbers?

**Solution:**

Multiples of 90 are 90,180,270,360,450,540 etc.

Given,

The multiple lies between 200 and 500 i.e. 270,360,450.

Also given the digits at tens place and hundreds place are consecutive numbers.

Hence, 450 is the number that lies between 200 and 500, and the digits at tens place and hundreds place are consecutive numbers.

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