Chapter - 3 - EXERCISE 3.5 - Playing with Numbers (NCERT Class 6th Maths Solution)

1. Which of the following statements are true? 
(a) If a number is divisible by 3, it must be divisible by 9. 
(b) If a number is divisible by 9, it must be divisible by 3. 
(c) A number is divisible by 18, if it is divisible by both 3 and 6. 
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90. 
(e) If two numbers are co-primes, at least one of them must be prime. 
(f) All numbers which are divisible by 4 must also be divisible by 8. 
(g) All numbers which are divisible by 8 must also be divisible by 4. 
(h) If a number exactly divides two numbers separately, it must exactly divide their sum. 
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.

Solution/Explanation:
(a) If a number is divisible by 3, it must be divisible by 9. [False]
(b) If a number is divisible by 9, it must be divisible by 3.[True]
(c) A number is divisible by 18, if it is divisible by both 3 and 6.[True]
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.[True]
(e) If two numbers are co-primes, at least one of them must be prime.[False]
(f) All numbers which are divisible by 4 must also be divisible by 8.[False]
(g) All numbers which are divisible by 8 must also be divisible by 4.[True]
(h) If a number exactly divides two numbers separately, it must exactly divide                 their sum.[True]
(i)If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.[False]


2. Here are two different factor trees for 60. Write the missing numbers. 

Solution/Explanation:

We can find the prime factors of a number by the factor tree method or by division method.

Here in question     

a) As factors of 6 = 2 × 3

Factors of 10 = 5 × 2

b) factors of 60 = 30 × 2

Factors of 30 = 10 × 3

Factors of 10= 5 × 2

3. Which factors are not included in the prime factorisation of a composite number?

Solution/Explanation:

Composite factors are not included in the prime factorisation of a composite number.

Since composite factors need to be broken further to prime number to get prime factorization.

4. Write the greatest 4-digit number and express it in terms of its prime factors.

Solution/Explanation:

Greatest four-digit number$=9999$

$9999=3\times 3\times 11\times 101$.

5. Write the smallest 5-digit number and express it in the form of its prime factors.

     

Solution/Explanation:

Smallest five digit number is 10000.

Prime factors of 10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5

6. Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.

Solution/Explanation:

On checking with 3,5,7,9,11, 13 we find 13 is a factor
1729 = 13 * 133  again checking with 17, 13, 19, we find
       = 13 * 19 * 7  =  7 * 13 * 19
we notice that the prime factors have a difference of 6. SO they are in arithmatic progression.

7. The product of three consecutive numbers is always divisible 
      by 6. Verify this statement with the help of some examples.

Solution/Explanation:

Step-by-step explanation: example

1) 2*3*4=24

2)5*6*7=210

3)1*2*3=6

so, it can be seen that all the numbers are divisible by 6

8. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples. 

Solution/Explanation:

3+ 5 = 8 and 8 is divisible by 4.
5 + 7 = 12 and 12 is divisible by 4.
7+ 9 = 16 and 16 is divisible by 4.
9 + 11 = 20 and 20 is divisible by 4.

9. In which of the following expressions, prime factorisation has     been done? 

(a) 24 = 2 × 3 × 4                            (b) 56 = 7 × 2 × 2 × 2 

(c) 70 = 2 × 5 × 7                            (d) 54 = 2 × 3 × 9 

Solution/Explanation:

(a) 24 = 2 × 3 × 4 Since 4 is composite, prime factorisation has not been done. 

(b) 56 = 7 × 2 × 2 × 2 Since ail the factors are prime, prime factorisation has been         done. 

(c) 70 = 2 × 5 × 7 Since all the factors are prime, prime factorisation has been done. (d) 54 = 2 × 3 × 9 Since 9 is composite, prime factorisation has not been done.

10. Determine if 25110 is divisible by 45. [Hint : 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].

Solution/Explanation:

We will check the divisibility of 25110 by 3 and 5 .

Divisibility of 25110 by 3

if a number is divisible by 3 then sum of digits of that number must be divisible by 3.

2+5+1+1+0 = 9 
⇒ 25510 is divisible by 3.

Divisibility by 5
If a number ends with 0 or 5 , then that number is divisible by 5.

25110 ends with 0 
⇒25110 is divisible by 5.

If 25110 is divisible by 3 and 5 , then it is divisible by 45 also.


11. 18 is divisible by both 2 and 3. It is also divisible by 2 × 3 = 6. Similarly, a number is divisible by both 4 and 6. Can we say that the number must also be divisible by 4 × 6 = 24? If not, give an example to justify your answer.

Solution/Explanation:

No,  It is not necessary because $12$ and $36$ are divisible by $4$ and $6$ both, but are not divisible by $24$.

12. I am the smallest number, having four different prime factors. Can you find me?

Solution/Explanation:
First four distinct prime numbers = 2,3,5,7

2×3×5×7 = 210

210 is the required number.