Question 2
Co-prime art
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?
We will look at how many pegs are on each circle and how far apart the thread jumps.
Step 1 — First Diagram Details
Look at the first diagram. It has numbers from 1 to 12. So, there are 12 pegs. The blue thread starts at peg 12. It goes to peg 4. To go from 12 to 4, we count 4 steps. (From 12 to 1, then 1 to 2, 2 to 3, 3 to 4). The thread-gap is 4. The thread does not touch every peg. It only touches pegs 12, 4, and 8. Let us find common factors of 12 and 4. Factors of 12 are 1, 2, 3, 4, 6, 12. Factors of 4 are 1, 2, 4. The biggest common factor is 4.
Step 2 — Second Diagram Details
Look at the second diagram. It has numbers from 1 to 13. So, there are 13 pegs. The red thread starts at peg 13. It goes to peg 3. To go from 13 to 3, we count 3 steps. (From 13 to 1, then 1 to 2, 2 to 3). The thread-gap is 3. The thread touches every peg. Let us find common factors of 13 and 3. Factors of 13 are 1, 13. Factors of 3 are 1, 3. The biggest common factor is 1.
Step 3 — Third Diagram Details
Look at the third diagram. It has numbers from 1 to 16. So, there are 16 pegs. The purple thread starts at peg 16. It goes to peg 6. To go from 16 to 6, we count 6 steps. (From 16 to 1, then 1 to 2, 2 to 3, 3 to 4, 4 to 5, 5 to 6). The thread-gap is 6. The thread does not touch every peg. It misses pegs like 1, 3, 5. Let us find common factors of 16 and 6. Factors of 16 are 1, 2, 4, 8, 16. Factors of 6 are 1, 2, 3, 6. The biggest common factor is 2.
Step 4 — Fourth Diagram Details
Look at the fourth diagram. It has numbers from 1 to 24. So, there are 24 pegs. The green thread starts at peg 24. It goes to peg 6. To go from 24 to 6, we count 6 steps. (From 24 to 1, then 1 to 2, 2 to 3, 3 to 4, 4 to 5, 5 to 6). The thread-gap is 6. The thread does not touch every peg. It only touches pegs 24, 6, 12, and 18. Let us find common factors of 24 and 6. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Factors of 6 are 1, 2, 3, 6. The biggest common factor is 6.
Step 5 — Co-prime Relationship
Let us look at our findings. For the first diagram, pegs are 12, gap is 4. The biggest common factor of 12 and 4 is 4. The thread did not touch every peg.
For the second diagram, pegs are 13, gap is 3. The biggest common factor of 13 and 3 is 1. The thread touched every peg.
For the third diagram, pegs are 16, gap is 6. The biggest common factor of 16 and 6 is 2. The thread did not touch every peg.
For the fourth diagram, pegs are 24, gap is 6. The biggest common factor of 24 and 6 is 6. The thread did not touch every peg.
We can see a pattern. If the biggest common factor of the number of pegs and the thread-gap is 1, the thread touches every peg. If the biggest common factor is more than 1, the thread does not touch every peg. Numbers with a biggest common factor of 1 are called co-prime numbers. So, if the number of pegs and the thread-gap are co-prime, the thread touches every peg. If they are not co-prime, the thread does not touch every peg.
Answer
(i) The third diagram has 16 pegs with a thread-gap of 6. (ii) The fourth diagram has 24 pegs with a thread-gap of 6. (iii) The thread touches every peg if the number of pegs and the thread-gap are co-prime.
More questions in A
Let us now play the 'idli-vada' game with different pairs of numbers:
a. 2 and 5, b. 3 and 7, c. 4 and 6.
We will say 'idli' for multiples of the smaller number, 'vada' for multiples of the larger number and 'idli-vada' for common multiples. Draw a figure similar to Fig. 5.1 if the game is played up to 60.
Co-prime art
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
Below are some boxes with four numbers in each box. Within each box try to say how each number is special compared to the rest. Share with your classmates and find out who else gave the same reasons as you did. Did anyone give different reasons that may not have occurred to you?
A prime puzzle
The figure on the left shows the puzzle. The figure on the right shows the solution of the puzzle. Think what the rules can be to solve the puzzle.
Rules
Fill the grid with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.
Q. Solve the prime puzzles by filling the grids with prime numbers only so that the product of each row is the number to the right of the row and the product of each column is the number below the column.