NCERT Solutions for Class 6th Maths Chapter 5 In-text Questions — Co-prime numbers for safekeeping treasures

Book page 115 to 117 Updated on2026-09-19

Q1.

Where should Grumpy place the treasures so that Jumpy cannot reach both the treasures? Check if these pairs are safe: a. 15 and 39 b. 4 and 15 c. 18 and 29 d. 20 and 55

Answer

Remember the new rule — a jump size of 1 is not allowed. So a pair is safe only if the two numbers have no common factor other than 1.

Pair	Factors	Common factors	Safe?
a. 15 and 39	15 → 1, 3, 5, 15 39 → 1, 3, 13, 39	1, 3	Not safe — jump 3 reaches 15 and 39
b. 4 and 15	4 → 1, 2, 4 15 → 1, 3, 5, 15	only 1	Safe ✔
c. 18 and 29	18 → 1, 2, 3, 6, 9, 18 29 → 1, 29	only 1	Safe ✔
d. 20 and 55	20 → 1, 2, 4, 5, 10, 20 55 → 1, 5, 11, 55	1, 5	Not safe — jump 5 reaches 20 and 55

Safe pairs: (4, 15) and (18, 29).

Why 29 makes a pair safe so easily: 29 is a prime, so its only factors are 1 and 29. Since 29 does not divide 18, the only common factor left is 1.

Q2.

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

Answer

Pair	Prime factors	Common prime factor?	Co-prime?
a. 18 and 35	18 = 2 × 3 × 3 35 = 5 × 7	none	Yes ✔
b. 15 and 37	15 = 3 × 5 37 is prime	none	Yes ✔
c. 30 and 415	30 = 2 × 3 × 5 415 = 5 × 83	5	No
d. 17 and 69	17 is prime 69 = 3 × 23	none	Yes ✔
e. 81 and 18	81 = 3 × 3 × 3 × 3 18 = 2 × 3 × 3	3	No

Co-prime pairs: a (18 and 35), b (15 and 37) and d (17 and 69).

Tip: both numbers in a co-prime pair need not be prime. 18 and 35 are both composite, yet they are co-prime, because they are built from completely different primes.

Q3.

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?

Answer

Number pair	First common multiple	Product	Same or less?	Co-prime?
3 and 5	15	15	Same	Yes
3 and 7	21	21	Same	Yes
4 and 9	36	36	Same	Yes
4 and 6	12	24	Less	No (share 2)
3 and 6	6	18	Less	No (share 3)
6 and 15	30	90	Less	No (share 3)

The rule Anshu spotted:

If the two numbers are co-prime, the first common multiple is exactly their product.
If they share a common factor, the first common multiple is less than the product — in fact it is the product divided by the biggest common factor.

4 and 6 share 2 → first common multiple = 24 ÷ 2 = 12
6 and 15 share 3 → first common multiple = 90 ÷ 3 = 30
3 and 6 share 3 → first common multiple = 18 ÷ 3 = 6

Why: the product always is a common multiple. But when the two numbers share a factor, that factor gets counted twice in the product, so a smaller common multiple already exists. Co-prime numbers have nothing to share, so nothing smaller than the product can work.

Q4.

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.

The four thread-art pictures on page 116. In each one the pegs are numbered round the circle and the thread runs from peg to peg by a fixed thread-gap.

Answer

Read each picture by counting the pegs on the circle and the pegs the thread actually touches.

The four thread pictures of the book, redrawn. Notice how many pegs the thread reaches in each one.

Diagram	Pegs	Thread-gap	Biggest common factor	Pegs the thread touches	Shape you see
1st	12	4	4	12 ÷ 4 = 3	a triangle
2nd	13	3	1	13 ÷ 1 = 13 (all)	a 13-pointed star
3rd	16	6	2	16 ÷ 2 = 8	an 8-pointed star
4th	24	6	6	24 ÷ 6 = 4	a square

1st: 12 → 4 → 8 → back to 12. Only pegs 4, 8, 12 are used.
4th: 24 → 6 → 12 → 18 → back to 24. Only pegs 6, 12, 18, 24 are used.

The rule: number of pegs touched = (number of pegs) ÷ (biggest common factor of the two numbers). When that common factor is 1 — that is, when the numbers are co-prime — every single peg is touched.

Q5.

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?

Answer

Yes, exactly. The thread reaches every peg if and only if the number of pegs and the thread-gap are co-prime.

Pegs and gap	Co-prime?	Does the thread reach every peg?
12 and 4	No (share 4)	No — only 3 of the 12 pegs
13 and 3	Yes	Yes — all 13 pegs
16 and 6	No (share 2)	No — only 8 of the 16 pegs
24 and 6	No (share 6)	No — only 4 of the 24 pegs

Why it happens: the thread lands on peg numbers that are multiples of the gap, counted round and round the circle. It comes back to the starting peg for the first time at the first common multiple of the two numbers. If they are co-prime, that first common multiple is the full product — so the thread has to go all the way round the circle (gap) times, visiting every peg. If they share a factor, it returns early and misses the rest.

Try This: take 13 pegs with a gap of 5, or 11 pegs with a gap of 4 — both are co-prime pairs, so both will give a star that touches every peg.

Q6.

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

Answer

First decide, for each pair, how many pegs the thread will touch. Then draw.

The four pictures asked for. Only (a) and (c) miss some pegs — in (b) and (d) the numbers are co-prime, so every peg is used.

Part	Pegs	Gap	Co-prime?	Pegs touched	Picture
a.	15	10	No (share 5)	15 ÷ 5 = 3	a triangle on pegs 10, 5, 15
b.	10	7	Yes	all 10	a 10-pointed star
c.	14	6	No (share 2)	14 ÷ 2 = 7	a 7-pointed star on the even pegs
d.	8	3	Yes	all 8	an 8-pointed star

a. 15 → 10 → 5 → back to 15 (only three pegs)
b. 10 → 7 → 4 → 1 → 8 → 5 → 2 → 9 → 6 → 3 → back to 10 (all ten pegs)
c. 14 → 6 → 12 → 4 → 10 → 2 → 8 → back to 14 (seven pegs, all even)
d. 8 → 3 → 6 → 1 → 4 → 7 → 2 → 5 → back to 8 (all eight pegs)

Tip for drawing: mark the pegs evenly on a circle with a compass, number them, then join peg to peg counting the gap each time. Stop when you come back to where you started.

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