Common Multiples and Common Factors
Children sit in a circle and play a game of numbers.
One of the children starts by saying 1. The second player says 2, and so on. But when it is the turn of 3, 6, 9, … (multiples of 3), the player should say 'idli' instead of the number. When it is the turn of 5, 10, … (multiples of 5), the player should say 'vada' instead of the number. When a number is both a multiple of 3 and a multiple of 5, the player should say 'idli-vada'! If a player makes any mistake, they are out.
The game continues in rounds till only one person remains.
For which numbers should the players say 'idli' instead of saying the number? These would be 3, 6, 9,
For which numbers should the players say 'vada'? These would be 5,
Which is the first number for which the players should say, 'idli-vada'? It is
1.
1. At what number is 'idli-vada' said for the 10th time?
Answer: We know that 'idli-vada' is said when a number is a multiple of both 3 and 5, which means we are looking for multiples of 15 (since 15 is the smallest number that is both a multiple of 3 and 5). So, the multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, ... The 10th multiple of 15 is
2.(a)
2. If the game is played for the numbers 1 to 90, find out:
Answer If the game is played from numbers 1 to 90:
a. How many times would the children say ‘idli’ (including the times they say ‘idlivada’)?
Answer
• The children will say ‘idli’ for multiples of 3 (like 3,
• The multiples of 3 from 1 to 90 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42,45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90
That’s 30 multiples of 3. However, 'idli-vada' is said for multiples of 15, so we must subtract those:
• The multiples of 15 from 1 to 90 are: 15, 30, 45, 60, 75, 90 (6 numbers). So, the number of times 'idli' is said is 30 - 6 =
2.(b)
b. How many times would the children say 'vada' (including the times they say 'idli-vada')?
Answer
• The children will say ‘vada’ for multiples of 5 (like 5, 10, 15, 20, etc.).
• The multiples of 5 from 1 to 90 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65,70, 75, 80, 85, 90 That’s 18 multiples of 5. Again, we must subtract the multiples of 15 (where 'idli-vada' is said instead of ‘vada’):
• The multiples of 15 from 1 to 90 are: 15, 30, 45, 60, 75, 90 (6 numbers).
So, the number of times 'vada' is said is 18 - 6 = 12 times.
2.(c)
c. How many times would the children say 'idli-vada'?
Answer As mentioned earlier, 'idli-vada' is said for multiples of 15. The multiples of 15 from 1 to 90 are: 15, 30, 45, 60, 75, 90 (6 times).
3.(a)
3. What if the game was played till 900? How would your answers change? Answer
If the game was played till 900, here’s how the answers would change:
a. How many times would the children say ‘idli’? Answer
• The multiples of 3 from 1 to 900 are: 3, 6, 9, ..., 900. There are
• The multiples of 15 from 1 to 900 are: 15, 30, 45, ..., 900. There are
So, the number of times ‘idli’ is said is 300 - 60 =
3.(b)
b. How many times would the children say ‘vada’?
• The multiples of 5 from 1 to 900 are: 5, 10, 15, ..., 900. There are 180 multiples of 5.
• The multiples of 15 from 1 to 900 are: 15, 30, 45, ..., 900. There are 60 multiples of 15.
So, the number of times ‘vada’ is said is 180 - 60 = 120 times.
3.(c)
c. How many times would the children say ‘idli-vada’?
Answer The multiples of 15 from 1 to 900 are: 15, 30, 45, ..., 900. So, the number of times ‘idli-vada’ is said is 60 times.
4.
4. Is this figure somehow related to the 'idli-vada' game?
Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.
Answer If you imagine the game being played till 30, and draw a pattern, you'll see that multiples of 3 and 5 (like 15) are intersections, which represent the numbers where ‘idlivada’ is said. This shows that the numbers where both conditions overlap are
Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.
Let us now play the 'idli-vada' game with different pairs of numbers:
We will say 'idli' for multiples of the smaller number, 'vada' for multiples of the larger number and 'idli-vada' for common multiples.
a.
a. 2 and 5
Answer:
When playing with 2 and 5:
'idli' for multiples of 2:
, , , , , , , , , , , , , , , , , , , , , , , , , , , , , 'vada' for multiples of 5:
, , , , , , , , , , , 'idli-vada' for common multiples (multiples of 10):
, , , , ,
From 1 to 60:
'idli' said:
times 'vada' said:
times 'idli-vada' said:
times
b.
b. 3 and 7
Answer:
When playing with 3 and 7:
'idli' for multiples of 3:
, , , , , , , , , , , , , , , , , 'vada' for multiples of 7:
, , , , , 'idli-vada' for common multiples (multiples of 21):
,
From 1 to 60:
'idli' said:
times 'vada' said:
times 'idli-vada' said:
times
c.
Answer:
When playing with 4 and 6:
'idli' for multiples of 4:
, , , , , , , , , 'vada' for multiples of 6:
, , , , 'idli-vada' for common multiples (multiples of 12):
, , , ,
From 1 to 60:
'idli' said:
times 'vada' said:
times 'idli-vada' said:
times
2.
Answer:
3.
3. Which of the following could be the other number: 2, 3, 5, 8, 10?
Answer:
This question appears incomplete as it doesn't specify what "the other number" refers to. However, if we're looking at which numbers could work well in the idli-vada game:
: Works well with any odd number to create interesting patterns : The original game uses 3, works with any number not divisible by 3 : The original game uses 5, works with any number not divisible by 5 : Could work, creates multiples every 8 number : Could work but might be too spread out for an engaging game
Best combinations would be:
2 and 3 (common multiples: 6, 12, 18, ...)
2 and 5 (common multiples: 10, 20, 30, ...)
3 and 5 (common multiples: 15, 30, 45, ...)
The key is choosing numbers that are