**Playing With Numbers Class 6 Ex. 3.3**

**Ncert Class 6 Math Free Solution.**

**Exercise 3.3**

**Question 1 :- Using divisibility tests, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10 ; by 11 (say, yes or no): **

**Number Divisible by**

** 2 3 4 5 6 8 9 10 11**

**128 Yes No Yes No No Yes No No No**

**990 ___ ___ ___ ___ ___ ___ ___ ___ ____**

**1586 ___ ___ ___ ___ ____ ___ ____ ___ ____**

**275 ___ ___ ___ ___ ____ ____ ___ ____ ____**

**6686 ___ ____ ____ ____ ____ ____ ____ ____ ____**

**639210 ____ ____ ____ ____ ____ _____ ____ ____ _____**

**429714 ____ ____ ____ ____ ____ ____ ____ ____ ____**

**2856 ____ ____ ____ ____ ____ ___ _____ ____ _____**

**3060 ____ ____ ____ ____ ____ ____ ____ ____ _____**

**406839 ____ ____ ____ ____ ____ ____ ____ ____ _____**

**Answer 1:-**

**Number Divisible by**

** 2 3 4 5 6 8 9 10 11**

**128 Yes No Yes No No Yes No No No**

**990 ** Yes** **Yes No Yes Yes No Yes Yes Yes

**1586 **No No No No No No No No No

**275 **Yes No No Yes No No No No Yes

**6686 **Yes No No No No No No No No

**639210 **Yes Yes No Yes Yes No No Yes Yes

**429714 **Yes Yes No No Yes No Yes No No

**2856 **Yes Yes Yes No Yes Yes No No No

**3060 **Yes Yes Yes Yes Yes No Yes Yes No

**406839 **No Yes No No No No No No No

**Question 2:- Using divisibility test, determine which of the following numbers are divisibly by 4; by 8:
(a) 572 (b) 726352 (c) 5500 (d) 6000 (e) 12159**

**(f) 14560 (g) 21084 (h) 31795072 (i) 1700 (j) 2150**

**Answer 2:-**

(a) 572

(i) Divisible by 4 as its last two digits are divisible by 4.

(ii) Not divisible by 8 as its last three digits are not divisible by 8.

(b) 726352

(i) Divisible by 4 as its last two digits are divisible by 4.

(ii) Divisible by 8 as its last three digits are divisible by 8.

(c) 5500

(i) Divisible by 4 as its last two digits are divisible by 4.

(ii) Not divisible by 8 as its last three digits are not divisible by 8.

(d) 6000

(i) Divisible by 4 as its last two digits are 0.

(ii) Divisible by 8 as its last three digits are 0.

(e) 12159

(i) Not divisible by 4 as its last two digits are odd numbers.

(ii) Not divisible by 8 as its last three digits are odd numbers.

(f) 14560

(i) Divisible by 4 as its last two digits are divisible by 4.

(ii) Divisible by 8 as its last three digits are divisible by 8.

(g) 21084

(i) Divisible by 4 as its last two digits are divisible by 4.

(ii) Not divisible by 8 as its last three digits are not divisible by 8.

(h) 31795072

(i) Divisible by 4 as its last two digits are divisible by 4.

(ii) Divisible by 8 as its last three digits are divisible by 8.

(i) 1700.

(ii) Not divisible by 8 as its last three digits are not divisible by 8.

(j) 2150

(i) Not divisible by 4 as its last two digits are not divisible by 4.

(ii) Not divisible by 8 as its last three digits are not divisible by 8.

Divisible by 4 : (a), (b), (c), (d), (f), (g), (h), (i)

Divisible by 8 : (b), (d), (f), (h)

**Question 3 :-Using divisibility tests, determine which of the following numbers are divisible by 6:**

**(a) 297144**

**(b) 1258**

**(c) 4335**

**(d) 61233**

**(e) 901352**

**(f) 438750**

**(g) 1790184**

**(h) 12583**

**(i) 639210**

**(j) 17852**

**Solution 3:-**

If a number is also divisible by both 2 and 3, then we know that it is also divisible by 6.

(a) 297144

The number 297144 has an even digit in the place of a one.

So, it is divisible by 2.

The sum of all the digits of 297144 = 2 + 9 + 7 + 1 + 4 + 4 = 27

which is divisible by 3.

As a result, the number 297144 is divisible by 6.

(b) 1258

The given number 1258 has even digit at its ones place.

So, it is divisible by 2.

The sum of all digits of 1258 = l + 2 + 5 + 8 = 16 which is not divisible by 3.

As a result, the number 1258 is not divisible by 6.

(c) 4335

In the given number, the digit at position one is not even.

So, it is not divisible by 2.

The sum of all the digits of 4335 = 4 + 3 + 3 + 5 = 15 which is divisible by 3.

Given that the number 4335 cannot be divided by both 2 and 3, it cannot be divided by 6.

(d) 61233

In the given number, the digit at position one is not even.

So, it is not divisible by 2.

The sum of the digits of the given number 61233 = 6 + 1 + 2 + 3 + 3 = 15 which is divisible by 3.

The number supplied cannot be divided by both 2 and 3, hence it cannot be divided by 6.

(e) 901352

In the given number, the digit at position one is even.

So, it is divisible by 2.

The sum of all the digits of the given number 901352 = 9 + 0 + 1 + 3 + 5 + 2 = 20 which is not divisible by 3.

The supplied number cannot be divided by both 2 and 3, hence it cannot be divided by 6.

(f) 438750

Ones place digit of the given number is 0. So, it is divisible by 2.

The sum of the digits =4 + 3 + 8 + 7 + 5 + 0 = 27 which is divisible by 3.

Therefore, the supplied number can be divided by 6.

(g) 1790184

In the given number, the digit at position one is even.

So, it is divisible by 2.

The sum of the digits = 1 + 7 + 9 + 0 + 1 + 8 + 4 = 30 which is divisible by 3.

Therefore, the supplied number can be divided by 6.

(h) 12583

The supplied number has an odd digit in the place of one.

So, it is not divisible by 2.

The sum of all the digits = 1 + 2 + 5 + 8 + 3 = 19 which is not divisible by 3.

Therefore, the supplied number can be divided by 6.

(i) 639210

The digit at the place of ones is 0.

So, it is divisible by 2.

The sum of the digits = 6 + 3 + 9 + 2 + 1 + 0 = 21 which is divisible by 3.

Therefore, the supplied number can be divided by 6.

(j) 17852

In the given number, the digit at position one is even.

So, it is divisible by 2.

The sum of the digits = 1 + 7 + 8 + 5 + 2 = 23 which is not divisible by 3.

Therefore, the supplied number can be divided by 6.

**Question 4 :- Using divisibility tests, determine which of the following numbers are divisible by 11:**

**(a) 5445**

**(b) 10824**

**(c) 7138965**

**(d) 70169308**

**(e) 10000001.**

**Solution 4:-**

If the difference between the sum of the digits in the odd places (from the right) and the sum of the digits in the even places (from the right) of the number is either 0 or divisible by 11, then the number is divisible by 11.

(a) 5445

Sum of the digits at odd places = 5 + 4 = 9

Sum of the digits at even places = 4 + 5 = 9

Difference = 9 – 9 = 0

The supplied number can therefore be divided by 11.

(b) 10824

Sum of the digits at odd places = 4 + 8 + 1 = 13

Sum of the digits at even places = 2 + 0 = 2

Difference = 13 – 2 = 11

which is divisible by 11.

The supplied number can therefore be divided by 11.

(c) 7138965

Sum of the digits at odd places = 5 + 9 + 3 + 7 = 24

Sum of the digits at even places = 6 + 8 + 1 = 15

Difference = 24 – 15 = 9

which is not divisible by 11.

The supplied number can therefore be divided by 11.

(d) 70169308

Sum of all the digits at odd places = 8 + 3 + 6 + 0 = 17

Sum of all the digits at even places = 0 + 9 + 1 + 7 = 17

Difference = 17-17 = 0

The supplied number can therefore be divided by 11.

(e) 10000001

Sum of all the digits at odd places = 1 + 0 + 0 + 0 = 1

Sum of all the digits at even places = 0 + 0 + 0 + 1 = 1

Difference = 1 – 1 = 0

Hence, the given number is divisible by 11.

**Question 5 :- Write the smallest digit and the greatest digit in the blank space of each of the following numbers so that the number formed is divisible by 3.**

**(a) ____ 6724**

**(b) 4765 ____ 2**

**Solution 5:-**

If the sum of a number’s digits is also divisible by three, then we know the number is likewise divisible by three.

(a) ___ 6724

Sum of the digits = 6 + 7 + 2 + 4 = 19

Blank space = 2, is the least digit that can be used.

Then the sum = 19 + 2 = 21 which is divisible by 3.

The highest digit that can be used in the empty slot is 8.

Then, the sum = 19 + 8 = 27 which is divisible by 3

Consequently, the necessary digits are 2 and 8.

(b) 4765 ____ 2.

Sum of digits = 4 + 7 + 6 + 5 + 2 = 24

The smallest numerals that can go in the empty space are 0

Then, sum = 24 + 0 = 24

which is divisible by 3.

The highest digit that can be entered into a space is 9.

Then, the sum = 24 + 9 = 33 which is divisible by 3.

Consequently, the necessary digits are 0 and 9.

**Question 6. Write a digit in the blank space of each of the following numbers so that the numbers formed is divisible by 11.**

**(a) 92 ___ 389**

**(b) 8 ___ 9484**

**Solution 6 :-**If the difference between the sum of the digits in the odd and even locations should be either 0 or 11, then the number is divisible by 11.

(a) 92 ___ 389

Sum of the digits at odd places = 9 + 3 + 2 = 14

Sum of the digits at even places = 8 + ( ) + 9

Difference = 17 + ( ) – 14 = 11;

( ) = 8

Hence, the required number is **928389.**

(b) 8 ___ 9484

Sum of the digits at odd places = 4 + 4 + ( ) = 8 + ( )

Sum of the digits at even places = 8 + 9 + 8 = 25

∴ Difference = 25 – [8 + ( )]

= 25 – 8 – ( ) = 17 – ( )

For the given number to be divisible by 11

17 – 0 = 11

∴ 17 – 11 = 6

So, the missing digit = 6

Hence, the required number = **869484**.