CBSE Class 6 Mathematics Prime Time Assignment Set 01

Short Answer Questions

Question 1. Children play the idli-vada game with 3 and 5, counting from 1 up to 40.
(i) Write all the numbers for which the players must say ‘idli-vada’.
(ii) What is the special name given to such numbers?
Answer:
‘Idli-vada’ is said at a number that is a multiple of 3 and also a multiple of 5.
(i) Up to 40 these numbers are 15 and 30.
(ii) They are called the common multiples of 3 and 5.
In simple words: Numbers that are multiples of both 3 and 5 are common multiples: 15 and 30 up to 40.

Teacher's Note:
1. The next one, 45, is beyond 40.
2. Use the term ‘common multiples’ exactly.

Question 2. Find all the common factors of 30 and 42. Also write the largest among them.
Answer:
Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.
Common factors: 1, 2, 3 and 6.
The largest is 6; check: 30 = 6 × 5 and 42 = 6 × 7.
In simple words: List both sets of factors and keep the shared ones; the biggest shared one is 6.

Teacher's Note:
1. Pair factors to avoid missing any.
2. Verify the largest one by dividing.

Question 3. Explain why the number 1 is called neither a prime number nor a composite number.
Answer:
A prime number has exactly two factors, 1 and itself.
A composite number has more than two factors.
The number 1 has only one factor, namely 1. It does not have two factors, and it does not have more than two.
So 1 is neither prime nor composite.
In simple words: 1 has just one factor, so it fits neither definition.

Teacher's Note:
1. State both definitions first.
2. Then show 1 fails each one.

Question 4. Look at the number 90.
(i) Write its prime factorisation.
(ii) Write all its prime factors.
Answer:
Break 90 into factors until only primes remain: 90 = 9 × 10 = 3 × 3 × 2 × 5.
(i) 90 = 2 × 3 × 3 × 5.
(ii) Its prime factors are 2, 3 and 5.
In simple words: 90 = 2 × 3 × 3 × 5, so its prime factors are 2, 3 and 5.

Teacher's Note:
1. The factorisation repeats 3; the list of prime factors does not.
2. Write primes in increasing order.

Question 5. Check whether 4328 is divisible by 2 and by 4. Give the reason you used in each case.
Answer:
By 2: the last digit is 8, an even digit, so 4328 is divisible by 2.
By 4: only the last two digits matter. They form 28 = 4 × 7, so 4328 is divisible by 4.
Check: 4328 = 4 × 1082.
In simple words: Even last digit gives 2; last two digits 28 being a multiple of 4 gives 4.

Teacher's Note:
1. Name the test you use in each case.
2. A quick check by division earns confidence.

Question 6. What are co-prime numbers? Give one pair of co-prime numbers in which neither number is a prime, and explain your choice.
Answer:
Two numbers are co-prime if they have no common factor other than 1.
A suitable pair is 8 and 9; both are composite.
Factors of 8: 1, 2, 4, 8. Factors of 9: 1, 3, 9.
The only shared factor is 1, so 8 and 9 are co-prime even though neither is prime.
In simple words: Co-prime numbers share only the factor 1, like 8 and 9.

Teacher's Note:
1. Co-prime is about the pair, not about each number.
2. Show both factor lists as proof.

Question 7. Find the smallest common multiple of 6 and 8. Then write the next two common multiples of 6 and 8.
Answer:
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …
Multiples of 8: 8, 16, 24, 32, 40, 48, …
The smallest common multiple is 24.
The next two are the next multiples of 24: 48 and 72.
In simple words: 6 and 8 first meet at 24, then at 48 and 72.

Teacher's Note:
1. All common multiples are multiples of the first one.
2. 6 × 8 = 48 is not the smallest here.

Question 8. What are twin primes? Write all the pairs of twin primes that lie between 10 and 35.
Answer:
Twin primes are two prime numbers whose difference is 2.
Primes between 10 and 35: 11, 13, 17, 19, 23, 29, 31.
Pairs differing by 2: (11, 13), (17, 19) and (29, 31).
In simple words: List the primes and pick pairs that are 2 apart.

Teacher's Note:
1. 23 has no twin here: 21 and 25 are composite.
2. List primes carefully before pairing.

Question 9. In the treasure game, Grumpy keeps two treasures, one on 18 and one on 45. Jumpy starts at 0 and must keep the same jump size. Find all the jump sizes with which Jumpy can reach both treasures, and say what such numbers are called.
Answer:
Jumpy lands on a number only when his jump size is a factor of it.
Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 45: 1, 3, 5, 9, 15, 45.
Jump sizes in both lists: 1, 3 and 9.
These are the common factors of 18 and 45.
In simple words: Jumps that reach both 18 and 45 are their common factors: 1, 3 and 9.

Teacher's Note:
1. Connect ‘jump size’ with ‘factor’.
2. Name the idea: common factors.

Question 10. While playing the idli-vada game, Anshu noticed that sometimes the first common multiple of the two numbers is exactly their product, and sometimes it is less than their product.
(i) Give one example of each kind.
(ii) Explain how this is connected to the pair being co-prime.
Answer:
(i) For 5 and 6 the first common multiple is 30, which equals 5 × 6. For 6 and 10 the first common multiple is also 30, but 6 × 10 = 60, so it is less than the product.
(ii) 5 and 6 are co-prime, so the first common multiple equals the product. 6 and 10 share the factor 2, so they are not co-prime and their first common multiple comes earlier than the product.
In simple words: Co-prime pairs meet first at their product; other pairs meet earlier.

Teacher's Note:
1. Pick small numbers to keep examples easy.
2. A shared factor lets the multiples meet sooner.

Question 11. Use prime factorisation to show that 45 and 56 are co-prime.
Answer:
45 = 3 × 3 × 5 and 56 = 2 × 2 × 2 × 7.
The prime factors of 45 are 3 and 5; those of 56 are 2 and 7. No prime is common.
Without a common prime factor there is no common factor bigger than 1, so 45 and 56 are co-prime.
In simple words: 45 and 56 share no prime, so their only common factor is 1.

Teacher's Note:
1. Write both factorisations clearly.
2. End with the word ‘co-prime’.

Question 12. Is 8120 divisible by 8? Is it divisible by 10? Use the divisibility tests of this chapter and write the reason in each case.
Answer:
By 8: the last three digits form 120 = 8 × 15, so 8120 is divisible by 8.
By 10: 8120 ends in 0, so it is divisible by 10.
Check: 8120 = 8 × 1015 = 10 × 812.
In simple words: Last three digits 120 work for 8, and the final 0 works for 10.

Teacher's Note:
1. 8 needs three digits; 10 needs one.
2. Write the test, then apply it.

Question 13. Write four consecutive composite numbers that lie between 20 and 30. For each one, write a factor other than 1 and itself to show that it is composite.
Answer:
Four such numbers are 24, 25, 26 and 27.
24 = 2 × 12, so 2 is a factor. 25 = 5 × 5, so 5 is a factor.
26 = 2 × 13, so 2 is a factor. 27 = 3 × 9, so 3 is a factor.
Each has more than two factors, so each is composite.
In simple words: 24, 25, 26, 27 each have an extra factor, so they are composite.

Teacher's Note:
1. The numbers between the primes 23 and 29 are all composite.
2. One extra factor per number is enough.

Question 14. Consider the number 2350.
(i) Which of 2, 5 and 10 are factors of 2350?
(ii) Show that 2350 is not divisible by 4.
Answer:
(i) 2350 ends in 0, so it is divisible by 2, 5 and 10. All three are factors.
(ii) For 4 only the last two digits matter. They form 50 = 4 × 12 + 2, which is not a multiple of 4. So 2350 is not divisible by 4.
In simple words: Ending in 0 gives 2, 5, 10; but 50 is not a multiple of 4.

Teacher's Note:
1. Divisible by 2 does not mean divisible by 4.
2. Show the remainder 2.

Question 15. For the number 144:
(i) write its prime factorisation;
(ii) use that factorisation to check whether 144 is divisible by 18.
Answer:
(i) 144 = 12 × 12 = (2 × 2 × 3) × (2 × 2 × 3), so 144 = 2 × 2 × 2 × 2 × 3 × 3.
(ii) 18 = 2 × 3 × 3. 144 has at least one 2 and two 3s, so the factorisation of 18 is fully inside that of 144.
Hence 144 is divisible by 18, and 144 = 18 × 8.
In simple words: 144 contains one 2 and two 3s, which is all that 18 needs.

Teacher's Note:
1. The left-over primes 2 × 2 × 2 = 8 give the quotient.
2. Count each prime carefully.

Question 16. Answer the following.
(i) Show that no even number other than 2 can be a prime number.
(ii) How many prime numbers are less than 20? List them.
Answer:
(i) Take any even number bigger than 2. It is divisible by 2, so it has at least the factors 1, 2 and itself. That is more than two factors, so it is composite. For 2 itself, the factor 2 is the number itself, so 2 has only two factors and is prime.
(ii) There are 8 primes less than 20: 2, 3, 5, 7, 11, 13, 17, 19.
In simple words: Even numbers above 2 have 2 as an extra factor; there are 8 primes below 20.

Teacher's Note:
1. Explain why 2 is the exception.
2. Count the list carefully.

Question 17. Show that two different prime numbers are always co-prime. Then explain, with one example, why two co-prime numbers need not be prime numbers themselves.
Answer:
Let p and q be two different primes. The factors of p are only 1 and p; the factors of q are only 1 and q.
Since p ≠ q, the only number in both lists is 1. So two different primes are always co-prime; for example, 7 and 13.
The reverse is not true. 8 and 9 are co-prime (only common factor 1), yet 8 = 2 × 2 × 2 and 9 = 3 × 3 are both composite.
Being co-prime is about the pair, not about each number by itself.
In simple words: Two different primes share only 1, but composite numbers like 8 and 9 can be co-prime too.

Teacher's Note:
1. Use letters to show the general case.
2. Give a clear counter-example for the second part.

Question 18. The prime factorisation of a number has two 2s, one 3 and one 5.
(i) Find the number.
(ii) Using prime factorisation, explain whether the number is divisible by 8.
Answer:
(i) The number is 2 × 2 × 3 × 5 = 60.
(ii) 8 = 2 × 2 × 2, so a multiple of 8 needs at least three 2s. 60 has only two 2s.
So 60 is not divisible by 8. (Check: 60 = 8 × 7 + 4.)
In simple words: 60 has only two 2s, but 8 needs three, so 8 does not divide 60.

Teacher's Note:
1. Count the 2s.
2. A quick division confirms the answer.

Question 19. Grumpy wants to keep two treasures on numbers between 20 and 30 so that Jumpy cannot reach both of them (a jump size of 1 is not allowed). Give one such safe pair and justify it using prime factorisation. Also give one pair from the same range that is not safe, and say why.
Answer:
A pair is safe exactly when the two numbers are co-prime, since then no jump size bigger than 1 reaches both.
Safe pair: 25 and 27. 25 = 5 × 5 and 27 = 3 × 3 × 3 share no prime, so their only common factor is 1. A jump of 1 is not allowed, so Jumpy cannot reach both.
Not safe: 24 and 28. 24 = 2 × 2 × 2 × 3 and 28 = 2 × 2 × 7 share the prime 2, so jump sizes 2 and 4 land on both.
In simple words: Choose co-prime numbers to keep treasures safe; numbers sharing a factor are not safe.

Teacher's Note:
1. Other safe pairs such as 21 and 22 are also correct.
2. Name the jump sizes that win for the unsafe pair.

Question 20. Four numbers are written in a box: 9, 15, 16, 23. For each number, give one reason why it is special when compared with the other three.
Answer:
9 is the only single-digit number (also the only multiple of 9).
15 is the only multiple of 5.
16 is the only even number (also the only multiple of 4).
23 is the only prime number; 9, 15 and 16 are composite.
Any other reason that picks out exactly one number is also correct.
In simple words: Each number has some property that none of the other three has.

Teacher's Note:
1. Check that the property fits only one number.
2. Divisibility and primeness give quick reasons.