# Ncert Class 6 Math Playing With Numbers Exercise 3.3

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                 ____    ____   ____   ____   ____  ____  ____   ____   _____

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

(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.

(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.

(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.