Page 143 - Ganit Kaushal

On which of the two numbers shall Dumpy place the treasure so that Jumpy cannot reach both treasures?

Will placing the treasures on 14 and 24 work?
The answer is No!  A jump size of 2 will allow Jumpy to land on both 14 and 24, as 2 is a common
factor of 14 and 24.
What about 5 and 24?
This time the answer is Yes! Jumpy cannot reach both using any jump size other than 1 and a jump
size of 1 is not allowed. So, Dumpy knows that the pair 5 and 24 is safe.
What do you understand by safe pairs?
Yes, you are right! They don’t have any common factor other than 1.
But isn’t that how we define co-primes!

Yes, you got it right; safe pairs are nothing but co-primes.
Ex 1. Check if these pairs are safe.
(a) 15 and 39 (b) 4 and 15
Sol. (a) 15 and 39 is not safe. Choosing a jump size of 3 will allow Jumpy to reach both numbers i.e., 15
and 39.
(b) 4 and 15 is safe.  Jumpy cannot reach both 4 and 15 using any jump size other than 1.
Ex 2. While playing the Dahi-Vada game with different number pairs, Aashu observed something interesting!
(a) Sometimes the first common multiple was the same as the product of the two numbers.
(b) At other times, the first common multiple was less than the product of the two numbers.
What do you understand by that? Find examples for each of the above. (NCERT)

Sol. (a) Such a number pair must be co-prime. Let us take 7 and 9. The first common multiple of 7 and 9
is 63, which is also the product of the two numbers.

(b) Such a number pair is not co-prime. Let us take 6 and 14. They are not co-prime as 2 is their common
factor other than 1. So, their first common multiple ‘42’ is less than their product, 6 × 14 = 84.

5.4 CO-PRIME ART

Let us take you on a journey of an interesting and intriguing thread art. Observe the following diagrams of thread
art. Diagram (i) has 13 pegs, and 12 13 15 16 1 12
the thread is tied to every third peg 1 14 2 11 1
(we say that the thread-gap is 3). 11 2 13 3 10 2
Diagram (ii) has 16 pegs, and the 10 3
thread is tied to every sixth peg (the 12 4 9 3
thread-gap is 6). Diagram (iii), has 9 4 11 5 8 4
12 pegs, and the thread-gap is 4. 8 5 10 6
After completing the art, do you 7 6 9 8 7 7 6 6 5
observe that, in diagram (i), the (i) (ii) (iii)
thread passes through every peg, whereas in the others it does not?
Is it related to the two numbers (the number of pegs and the thread-gap) being co-prime?
Certainly, they are related to being co-prime. You will get answers to all queries after observing and
analysing the three diagrams minutely.
In diagram (i), the number of pegs and the thread-gap, i.e., 13 and 3, are co-prime to each other. So, the thread
finally passes through every peg. ( Common factor = 1)

Prime Time 141