Page 137 - Ganit Kaushal

Ex. The HCF and LCM of two numbers
are 16 and 80, respectively. If one Remember
of the numbers is 32, find the 1. The HCF of the given numbers divides all of them.
other one. 2. The LCM of the given numbers is divisible by all of them.
Sol. Let the other number be a. 3. The HCF of given numbers is always a factor of their LCM.
By the above formula, 4. The HCF of given numbers ≤ all the numbers and the LCM
of given numbers ≥ all the numbers.
32 × a = 16 × 80 5. Common multiples of two or more given numbers are
⇒ a = (16 × 80) ÷ 32 = 40. multiples of their LCM.

Play and Learn [‘Dahi–Vada’ Game]

Now, it’s time to play a fun game ‘Dahi-Vada’, based on multiples and common multiples.

A few children sit together in a circle and play this game of numbers. The rules of the game are as follows:
1. One of the children starts by saying ‘1’. The second player says ‘2’, and so on.
2. But when it is the turn of 3, 6, 9, … (multiples of 3), the player should say ‘Dahi’ instead of the number.

3. When it is the turn of 5, 10, 15, … (multiples of 5), the player should say ‘Vada’ instead of the number.
4. When a number is both a multiple of 3 as well as a multiple of 5, the player should say ‘Dahi-
Vada’! If a player makes any mistake, the player is out.

5. The game continues in rounds till only one person remains and wins the game.
Now, answer a few questions associated with the game.
Can you tell the numbers when players say ‘Dahi’ instead of saying the number?
Yes, you are right. These would be multiples of 3, i.e., 3, 6, 9, 12, 18, and so on.
Can you tell the numbers when players say ‘Vada’ instead of saying the number?

Great! You are right again. These would be multiples of 5, i.e., 5, 10, 15, 20, and so on.
What is the first number for which the players should say, ‘Dahi-Vada’?
Absolutely right! It is 15, since it is the first multiple of both 3 and 5.
What are other such numbers for which the players should say, ‘Dahi-Vada’?
Great! Other common multiples of 3 and 5 are 30, 45, 60, and so on.

Did you notice that all these numbers are multiples of 15? Why is it so? Think.
Absolutely right! They are common multiples of 3 and 5. So, they must be multiples of the LCM (3,
5) = 15. (see above - Remember point no 5.)
Solved Examples

Ex 1. Find all multiples of 40 that lie between 310 and 410. (NCERT)
Sol. Multiples of 40 = 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 440…...
\ All multiples of 40 that lie between 310 and 410 are 320, 360, and 400.
Ex 2. Find the common factors of: (NCERT)
(a) 20 and 28 (b) 5, 15 and 25
Sol. (a) Factors of 20 are 1, 2, 4, 5, 10, and 20.

Factors of 28 are 1, 2, 4, 7, 14, and 28.
\ Common factors of 20 and 28 are 1, 2, and 4.

Prime Time 135