HCF and LCM: word problems

Key Notes:

Highest Common Factor (HCF):

  • The HCF of two or more numbers is the greatest number that divides each of them without leaving a remainder.
  • Example: The HCF of 12 and 18 is 6 because 6 is the largest number that can divide both 12 and 18.

Least Common Multiple (LCM):

  • The LCM of two or more numbers is the smallest number that is a multiple of each of them.
  • Example: The LCM of 4 and 5 is 20 because 20 is the smallest number that both 4 and 5 can divide evenly.

Steps to Solve Word Problems Involving HCF and LCM:

  1. Identify the Problem Type:
    • HCF Problems: Look for situations where you need to split something into equal parts, share equally, or find the maximum size of an item.
    • LCM Problems: Look for situations where you need to find a repeating event, synchronize schedules, or find the smallest time interval.
  2. Determine the Numbers Involved:
    • Carefully read the problem to identify the numbers you need to work with.
  3. Calculate the HCF or LCM:
    • Use prime factorization, division method, or listing multiples and factors to calculate the HCF or LCM.
  4. Interpret the Result:
    • Apply the HCF or LCM to answer the question in the word problem.

Examples of Word Problems:

  1. HCF Word Problem:
    • Problem: Three friends have 24, 36, and 60 candies. They want to divide them into the largest possible equal groups. How many candies will each group have?
    • Solution: Find the HCF of 24, 36, and 60.
      • Prime Factorization:
        • 24=2^3 ×3
        • 36= 2^2 × 3^2
        • 60=2^2×3×5
      • HCF: The common prime factors are 2^2 ×3=12.
      • Answer: Each group will have 12 candies.
  2. LCM Word Problem:
    • Problem: Two buses leave the station at the same time. One bus returns every 6 hours, and the other every 8 hours. After how many hours will they both return at the same time?
    • Solution: Find the LCM of 6 and 8.
      • Prime Factorization:
        • 6=2×3
        • 8= 2^3
      • LCM: 2^3 ×3=24
      • Answer: Both buses will return at the same time after 24 hours.
  3. Mixed HCF and LCM Problem:
    • Problem: A gardener has two types of plants. One type needs to be watered every 4 days, and the other every 6 days. If both were watered today, when will they both be watered again on the same day?
    • Solution: Find the LCM of 4 and 6.
      • Prime Factorization:
        • 4= 2^2
        • 6=2×3
      • LCM: 2^2 ×3=12
      • Answer: Both types of plants will be watered again on the same day in 12 days.

Learn with an example

Problem 1 :

A merchant has 120 ltrs of and 180 ltrs of two kinds of oil. He wants the sell oil by filling the two kinds of oil in tins of equal volumes. What is the greatest of such a tin.

Solution :

The given two quantities 120 and 180 can be divided by 10, 20,… exactly. That is, both the kinds of oils can be sold in tins of equal volume of 10, 20,… ltrs.

But, the target of the question is, the volume of oil filled in tins must be greatest.

So, we have to find the largest number which exactly divides 120 and 180.That is nothing but the H.C.F of (120, 180)

H.C.F of (120, 180)  =  60

The 1st kind 120 ltrs is sold in 2 tins of of volume 60 ltrs in each tin.

The 2nd kind 180 ltrs is sold in 3 tins of volume 60 ltrs in each tin.

Hence, the greatest volume of each tin is 60 ltrs.

Problem 2 :

Find the least number of square tiles by which the floor of a room of dimensions 16.58 m and 8.32 m can be covered completely.

Solution :

We require the least number of square tiles. So, each tile must be of maximum dimension.

To get the maximum dimension of the tile, we have to find the largest number which exactly divides 16.58 and 8.32. That is nothing but the H.C.F of (16.58, 8.32).

To convert meters into centimeters, we have to multiply by 100.

16.58 ⋅ 100  =  1658 cm

8.32 ⋅ 100  =  832 cm

H.C.F of (1658, 832)  =  2

Hence the side of the square tile is 2 cm

Required no. of tiles :

=  (Area of the floor) / (Area of a square tile)

=  (1658 ⋅ 832) / (2 ⋅ 2)

=  344,864

Hence, the least number of square tiles required is 344,864.

Problem 3 :

Lenin is preparing dinner plates. He has 12 pieces of chicken and 16 rolls. If he wants to make all the plates identical without any food left over, what is the greatest number of plates Lenin can prepare ?

Solution :

To make all the plates identical and find the greatest number of plates, we have to find the greatest number which can divide 12 and 16 exactly.

That is nothing but H.C.F of 12 and 16.

H.C.F of (12, 16)  =  4

That is, 12 pieces of chicken would be served in 4 plates at the rate of 3 pieces per plate.

And 16 rolls would be served in 4 plates at the rate of 4 rolls per plate.

In this way, each of the 4 plates would have 3 pieces of chicken and 4 rolls. And all the 4 plates would be identical.

Hence, the greatest number of plates Lenin can prepare is 4

Problem 4 :

The drama club meets in the school auditorium every 2 days, and the choir meets there every 5 days. If the groups are both meeting in the auditorium today, then how many days from now will they next have to share the auditorium ?

Solution :

If the drama club meets today, again they will meet after 2, 4, 6, 8, 10, 12…. days.

Like this, if the choir meets today, again they will meet after 5, 10, 15, 20 …. days.

From the explanation above, If both drama club and choir meet in the auditorium today, again, they will meet after 10 days.

And also, 10 is the L.C.M of (2, 5).

Hence, both the groups will share the auditorium after ten days.

Problem 5 :

John is printing orange and green forms. He notices that 3 orange forms fit on a page, and 5 green forms fit on a page. If John wants to print the exact same number of orange and green forms, what is the minimum number of each form that he could print ?

Solution :

The condition of the question is, the number of orange forms taken must be equal to the number of green forms taken.

Let us assume that he takes 10 orange and 10 green forms.

10 green forms can be fit exactly on 2 pages at 5 forms/page. But,10 orange forms can’t be fit exactly on any number of pages.

Because, 3 orange forms can be fit exactly on a page. In 10 orange forms, 9 forms can be fit exactly on 3 pages and 1 form will be remaining.

To get the number of forms in orange and green which can be fit exactly on some number of pages, we have to find L.C.M of (3,5). That is 15.

15 orange forms can be fit exactly on 5 pages at 3 forms/page.

15 green forms can be fit exactly on 3 pages at 5 forms/page.

Hence,the smallest number of each form could be printed is 15.

Let’s practice!🖊️