Q1. A supplier receives orders from 5 different buyers. Each buyer places their order only on a Monday. The first buyer places the order after every 2 weeks, the second buyer, after every 6 weeks, the third buyer, after every 8 weeks, the fourth buyer, every 4 weeks, and the fifth buyer, after every 3 weeks. It is known that on

January 1st, which was a Monday, each of these five buyers placed an order with the supplier. On how many occasions, in the same year, will these buyers place their orders together excluding the order placed on January 1st?

A 1

B 5

C 2

D 4

E 3

## EXPLANATION

C

The supplier receives his orders from the five buyers once every 2 weeks, once every 6 weeks, once every 8

weeks, once every 4 weeks, and once every 3 weeks.

The number of occasions where all the five buyers place the order on the same day is :

The LCM of the 5-time frames during which the 5 buyers place their orders :

Hence the LCM is :

(2, 6, 8, 4, 3).

= 24 weeks.

Once every 24 weeks, all five of them place the order simultaneously.

A year has 53 weeks in total :

Hence all five of them place the orders after 24 weeks, 48 weeks.

Q2. The Madhura Fruits Company is packing four types of fruits into boxes. There are126 oranges, 162 apples,198 guavas and 306 pears. The fruits must be packed in such a way that a given box must have only one type of fruit and must contain the same number of fruit units as any other box.

What is the minimum number of boxes that must be used?

A 21

B 18

C 44

D 42

E 36

## EXPLANATION

C

The number of oranges, apples, guavas, and pears = 126, 162, 198, and 306.

Each box must contain an equal number of fruits with only one type of fruit. The additional condition provided is that there should be a minimum number of boxes in total.

The distribution is possible in multiple ways in such a way that distribution in each box is placed in such that each box contains a certain number of fruits n which is a factor for all the four given number of fruits :

Arrangement of 1 fruit of one kind in a basket.

2 is a factor of 126, 162, 198, and 306. So we can place 2 fruits of a particular kind in a basket.

Since we were asked for the minimum number of boxes this is possible when a maximum number of fruits of a kind are placed in a box.

Hence each box must contain the Highest common factor for the four numbers :

The prime factorization for the four numbers :

126: 2 x 7 x 9,

162 : 2 x 9 x 9,

198 : 2 x 9 x 11,

306 : 2 x 9 x 17

The HCF is 18.

The number of boxes required for each :

7+9+11+17 = 44.

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