Page 142 - Ganit Kaushal

Co-prime Numbers

Two numbers are said to be co-prime if they have no common factors other than 1.
For example: 3 and 5; 5 and 7; 7 and 9; 8 and 15, etc.
Now, let’s play a game to understand factors, common factors, prime numbers and co-primes in a better way.

Jumping Jackpot (A Treasure Hunt Game)

Jumpy and Dumpy are friends. They play a game based on factors. The rules of the game are as follows:

1. Dumpy places a treasure on some number. For example, she may place it on 24.
2. Now, Jumpy has to choose a jump size. If she chooses 3, then she has to jump only on multiples
of 3, starting from 0.
3. Jumpy gets the treasure if she lands on the number where Dumpy has placed it.
Which jump sizes will get Jumpy to land on 24?
If she chooses 3, Jumpy will land as follows: 3 → 6 → 9 → 12 → 15 → 18 → 21 → 24. So, she lands
on 24 and will get the treasure. Other successful jump sizes are 2, 3, 4, 6, 8 and 12. (see figure)
What if Jumpy chooses jump sizes 1 and 24?
Yes, she will land on 24 and will get the treasure in this case as well.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

The numbers 1, 2, 3, 4, 6, 8, 12, and 24 all divide 24 exactly. These numbers are called factors or divisors of
24. So, we conclude that Jumpy gets the treasure only if:

Jump size = a divisor of 24

Dumpy now raises the difficulty level of the game. Instead of one treasure, Dumpy places two treasures on
two distinct numbers. Jumpy has to decide a jump size and maintain it. Jumpy receives the treasures only if
she lands on both numbers with the selected jump size. Jumpy starts from 0.

Suppose Dumpy has kept the treasures on 14 and 34. Jumpy chooses a jump size of 7.
Can you tell if Jumpy gets the treasures?
You are absolutely right! She does not get the treasure.  Starting from 0, she jumps to 7 → 14 → 21
→ 28 → 35 → 42 … She landed on 14 but did not reach 34.
Can you tell what jump size Jumpy should choose to get both the treasures?
You are right again! The jump sizes of 1 or 2 will allow Jumpy to land on both 14 and 34.

[ The factors of 14 are: 1, 2, 7, and 14 and the factors of 34 are: 1, 2, 17, and 34. Observe that 1, and
2 are common factors. So, the jump sizes of 1 or 2 will allow Jumpy to land on both 14 and 34.]

So, we conclude that Jumpy gets both the treasures if:
Jump size = a common factor of 14 and 34.

Now, Dumpy further increases the difficulty level of the game by adding one more rule: ‘A jump size of 1 is
not allowed’.

140 Mathematics-6