**Q) **National Art convention got registrations from students from all parts of the country, of which 60 are interested in music, 84 are interested in dance and 108 students are interested in handicrafts. For optimum cultural exchange, organisers wish to keep them in minimum number of groups such that each group consists of students interested in the same art form and the number of students in each group is the same. Find the number of students in each group. Find the number of groups in each art form. How many rooms are required if each group will be allotted a room?

**Ans: **

Let’s first understand the question.

We are given 3 art forms: Music, Dance and Handicrafts. Registrations received are 60, 84 and 108 respectively.

We need to form groups of students within each art form and each group should have equal number of students.

Clearly, this should be a number which is common across all the three art forms or we can say that this number should be a factor to all the three art form registration counts. **It means this number should be a common factor.**

Also it is given that the organizers want to keep the number of groups minimum. It means the the size of each group should be maximum, so that number of groups formed remain minimum. Let’s understand by example:

If there are 100 registrations and we make groups of 10 students each, then 10 groups will be formed. Similarly, if we make groups of 20 students each, then 5 groups will be formed and if we make groups of 50 students each, then 2 groups will be formed. Therefore, higher the size of group, less is the number of groups.

Since, it is given in the question that we need to make minimum number of groups, from above example, we clearly understand that we need to keep the largest group size and it should be common across three artforms. **It clearly means, we need to find HCF of all three registrations count.**

Let’s calculate the HCF of 60, 84 and 108 by prime factorisation method, we get:

60 = 2^{2} x 3 x 5

84 = 2^{2} x 3 x 7

108 = 2^{2} x 3^{3}

HCF = 2^{2} x 3 = 12

**Therefore, the number of students in each group will be 12.**

Next, let’s find the number of groups which will be formed within each of the art form registrations:

Number of groups formed for Music: = 5

Number of groups formed for Dance: = 7

Number of groups formed for Handicrafts: = 9

**Therefore, the number of groups formed in Music, Dance and Handicrafts artforms are 5, 7 and 9 respectively.**

Total number of rooms required = Number of rooms required for students of Music + Number of rooms required for students of Dance + Number of rooms required for students of Handicrafts

= 5 + 7 + 9 = 21

**Therefore, total 21 rooms will be required if each group is allotted a room.**