Chapter 1: Real Numbers

To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections - Section A and Section B of grade X. There are 32 students in section A and 36 students in section B.

1. What is the minimum number of books you will acquire for the class library, so that they can be distributed equally among students of Section A or Section B?
   (A) 144

   (B) 128

   (C) 288

   (D) 272
   Show Solution
   Solution: (C) 288
   We need to find the LCM of 32 and 36.
   $32 = 2^5$
   $36 = 2^2 \times 3^2$
   LCM $= 2^5 \times 3^2 = 32 \times 9 = 288$.
2. If the product of two positive integers is equal to the product of their HCF and LCM is true then, the HCF (32, 36) is?
   (A) 2

   (B) 4

   (C) 6

   (D) 8
   Show Solution
   Solution: (B) 4
   HCF $\times$ LCM = Product of numbers.
   HCF $\times 288 = 32 \times 36$
   HCF $= \frac{1152}{288} = 4$.

A seminar is being conducted by an Educational Organisation, where the participants will be educators of different subjects. The number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively.

1. In each room the same number of participants are to be seated and all of them being in the same subject, hence maximum number of participants that can accommodated in each room are?
   (A) 14

   (B) 12

   (C) 16

   (D) 18
   Show Solution
   Solution: (B) 12
   We need to find HCF(60, 84, 108).
   $60 = 2^2 \times 3 \times 5$
   $84 = 2^2 \times 3 \times 7$
   $108 = 2^2 \times 3^3$
   HCF $= 2^2 \times 3 = 12$.
2. What is the minimum number of rooms required during the event?
   (A) 11

   (B) 21

   (C) 29

   (D) 12
   Show Solution
   Solution: (B) 21
   Total participants $= 60 + 84 + 108 = 252$.
   Participants per room = 12 (HCF).
   Number of rooms $= 252 / 12 = 21$.

A mathematics exhibition is being conducted in your school and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience.

Observe the following factor tree: Number $x$ splits into 5 and $y$. $y$ splits into 2 and $z$. $z$ splits into 2 and 253. 253 splits into 11 and 23.

1. What will be the value of $z$?
   (A) 506

   (B) 253

   (C) 22

   (D) 46
   Show Solution
   Solution: (A) 506
   From the bottom up: $z = 2 \times 253 = 506$.
2. What will be the value of $x$?
   (A) 1012

   (B) 5060

   (C) 2530

   (D) 13915
   Show Solution
   Solution: (B) 5060
   $y = 2 \times z = 2 \times 506 = 1012$.
   $x = 5 \times y = 5 \times 1012 = 5060$.