Co-prime Numbers for Safekeeping Treasures
Which pairs are safe?
Let us go back to the treasure finding game. This time, treasures are kept on two numbers. Jumpy gets the treasures only if he is able to reach both the numbers with the same jump size. There is also a new rule—a jump size of 1 is not allowed.
Where should Grumpy place the treasures so that Jumpy cannot reach both the treasures?
Will placing the treasure on 12 and 26 work? No! If the jump size is chosen to be 2, then Jumpy will reach both 12 and 26.
What about 4 and 9? Jumpy cannot reach both using any jump size other than 1. So, Grumpy knows that the pair 4 and 9 is safe.
Check if these pairs are safe:
a. 15 and 39
b. 4 and 15
c. 18 and 29
d. 20 and 55
What is special about safe pairs? They don't have any common factor other than 1. Two numbers are said to be co-prime to each other if they have no common factor other than
Example: As 15 and 39 have 3 as a common factor, they are not co-prime. But 4 and 9 are co-prime.
Which of the following pairs of numbers are co-prime? a. 18 and 35
b. 15 and 37
c. 30 and 415
d. 17 and 69
e. 81 and 18
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While playing the 'idli-vada' game with different number pairs, Anshu observed something interesting!
1. Sometimes the first common multiple was the same as the product of the two numbers.
2. At other times the first common multiple was less than the product of the two numbers.
Find examples for each of the above. How is it related to the number pair being co-prime?
Observe the following thread art. The first diagram has 12 pegs and the thread is tied to every fourth peg (we say that the thread-gap is 4). The second diagram has 13 pegs and the thread-gap is 3. What about the other diagrams? Observe these pictures, share and discuss your findings in class.
In some diagrams, the thread is tied to every peg. In some, it is not. Is it related to the two numbers (the number of pegs and the thread-gap) being co-prime?
Make such pictures for the following:
a. 15 pegs, thread-gap of 10
b. 10 pegs, thread-gap of 7
c. 14 pegs, thread-gap of 6
d. 8 pegs, thread-gap of 3
Checking if two numbers are co-prime
Here's the corrected version with proper syntax:
Teacher: Are 56 and 63 co-prime?
Anshu and Guna: If they have a common factor other than 1, then they are not co-prime. Let us check.
Anshu: I can write 56 = 14 ×
Guna: Hold on. I can also write 56 = 7 × 8 and 63 = 9 × 7. We see that 7 is a factor of both numbers, so, they are not co-prime.
Clearly Guna is right, as 7 is a common factor.
But where did Anshu go wrong?
Writing 56 = 14 × 4 tells us that 14 and 4 are both factors of 56, but it does not tell all the factors of 56. The same holds for the factors of 63.
Try another example: 80 and 63. There are many ways to factorise both numbers.
80 = 40 × 2 = 20 × 4 = 10 × 8 = 16 × 5 = 1 ×
63 = 9 × 7 = 3 × 21 = 1 ×
We have written '1 × 80' to say that there may be more ways to factorise these numbers. But if we take any of the given factorisations, for example, 80 = 16 × 5 and 63 = 9 × 7, then there are no common factors. Can we conclude that 80 and 63 are co-prime? As Anshu's mistake above shows, we cannot conclude that as there may be other ways to factorise the numbers.
What this means is that we need a more systematic approach to check if two numbers are co-prime.