Chapter - 3 - EXERCISE 3.3 - Playing with Numbers (NCERT Class 6th Maths Solution)

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):

Solution/Explanation:

2. Using divisibility tests, determine which of the following numbers are divisible by 4; by 8:

(a) 572               (b) 726352            (c) 5500             (d) 6000  

(e) 12159           (f) 14560               (g) 21084           (h) 3179507

(i) 1700              (j) 2150

Solution/Explanation:

Divisibility by 4 and 8 is checked by the last 2 and 3 digits respectively. If the last two digit is is divisible by 4, then the number is divisible by 4. if the last three digits are divisible by 8,then the number is divisible by 8

(a) 572
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.

(b) 726352
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(c) 5500
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(d) 6000
Divisible by 4 as its last two digits are 0.
Divisible by 8 as its last three digits are 0.
(e) 12159
Not divisible by 4 and 8 as it is an odd number.
(f) 14560
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(g) 21084
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(h) 3179507
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.

3. Using divisibility tests, determine which of 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/Explanation:

(a) 297144 

Since the last digit Of the number is 4, it is divisible by 2. On adding all the digits of the number, the sum obtained is 27. Since 27 is divisible by 3, the given number is also divisible by 3. As the number is divisible by both 2 and 3, it is divisible by 6.

(b) 1258

Since the last digit of the number is 8, it is divisible by 2. On adding all the digits of the number, the sum obtained is 16. Since 16 is not divisible by 3, the given number is also not divisible by 3. As the number is not divisible by both 2 and 3, it is not divisible by 6.

(c) 4335

The last digit of the number is 5, which is not divisible by 2. Therefore, the given number is also not divisible by 2. On adding all the digits of the number, the sum obtained is 15. Since 15 is divisible by 3, the given number is also divisible by 3. As the number is not divisible by both 2 and 3, it is not divisible by 6.

(d) 61233

The last digit of the number is 3, which is not divisible by 2. Therefore, the given number is also not divisible by 2. On adding all the digits Of the number, the sum obtained is 15. Since 15 is divisible by 3, the given number is also divisible by 3. As the number is not divisible by both 2 and 3, it is not divisible by 6.

(e) 901352

Since the last digit of the number is 2, it is divisible by 2. On adding all the digits of the number, the sum obtained is 20. Since 20 is not divisible by 3, the given number is also not divisible by 3. As the number is not divisible by both 2 and 3, it is not divisible by 6.

(f) 438750

Since the last digit of the number is O, it is divisible by 2. On adding all the digits of the number, the sum obtained is 27. Since 27 is divisible by 3, the given number is also divisible by 3. As the number is divisible by both 2 and 3, it is divisible by 6.

(g) 1790184

Since the last digit of the number is 4, it is divisible by 2. On adding all the digits of the number, the sum obtained is 30. Since 30 is divisible by 3, the given number is also divisible by 3. As the number is divisible by both 2 and 3, it is divisible by 6.

(h) 12583

Since the last digit of the number is 3, it is not divisible by 2. On adding all the digits of the number, the sum obtained is 19. Since 19 is not divisible by 3, the given number is also not divisible by 3. As the number is not divisible by both 2 and 3, it is not divisible by 6.

(i) 639210

Since the last digit of the number is O, it is divisible by 2. On adding all the digits of the number, the sum Obtained is 21. Since 21 is divisible by 3, the given number is also divisible by 3. As the number is divisible by both 2 and 3, it is divisible by 6.

(j) 17852

Since the last digit of the number is 2, it is divisible by 2. On adding all the digits of the number, the sum obtained is 23. Since 23 is not divisible by 3, the given number is also not divisible by 3. As the number is not divisible by both 2 and 3, it is not divisible by 6.

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

(a) 5445                  (b) 10824                  (c) 7138965  

(d) 70169308          (e) 10000001            (f) 901153

Solution/Explanation:

(a) 5445:-  Sum of the given digits at odd places = 5 + 4 = 9 

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

Difference = 9 – 9 = 0 

As the difference between the sum of the digits at odd places and sum of the digits at even place is 0. Therefore, 5445 is divisible by 11. 

(b) 10824:-  Sum of the given digits at odd places = 4 + 8 + 1 = 13 

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

Difference = 13 – 2 = 11 

The difference between the sum of the digits at odd places and the sum of the digit at even places is 11. Which is divisible by 11. Therefore 10824 is divisible by 11. 

(c) 7138965:-  Sum of the given digits at odds places = 5 + 9 + 3 + 7 = 24 

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

Difference = 24 – 15 = 9. 

The difference between the sum of the digits at odd places and the sum of digits at even place is 9, Which is not divisible by 11. ∴ 7138965 is not divisible by 11. 

(d) 70169308:-  Sum of the digits at odd places = 8 + 3 + 6 + 0 = 17 

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

Difference = 17 – 17 = 0 

As the difference between the sum of the digits at odd places and the sum of the digits at even place is 0. Therefore, 70169308 is divisible by 11. 

(e) 10000001:-  Sum of the digits at odd places=1 1 

Sum of the digits even place = 1 Difference = 1 – 1 = 0 

As the difference the sum of the digits at odd places and the sum of the digits at even places is 0. therefore 10000001 is divisible by 11. 

(f) 901153:-  Sum of the digits at odd places = 3 + 1 + 0 = 4. 

Sum of the digits at even places = 5 + 1 + 9 = 15 

Difference = 15 – 4 = 11 

The difference between the sum of the digits at odd places and the sum of the digits at even places is 11, Which is divisible by 11. Therefore, 901153 is divisible by 11.

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/Explanation:

If the sum of the digits in a number is a multiple of 3 then the number will be divisible by 3.

(a) Given, number is _6724.

The sum of the remaining digits is given as,

6+7+2+4=19

Since the smallest multiple of 3 that comes after 19 is 21 and the greatest multiple of 3 according to given condition that comes after 19 is 27.

Smallest number  21−19=2

Greatest number  27−19=8

Thus, the smallest number is 2$2$ and the largest number is 8.

(b) The sum of the remaining digits is given as,

4+7+6+5+2=24

Since 24 is divisible by 3, so smallest number at blank space will be 0 and the greatest multiple of 3 that comes after 24 is 33.

Greatest number 33−24=9

Thus, the smallest number is 0 and the largest number is 9.

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

(a) 92 __ 389              (b) 8 __ 9484

Solution/Explanation: