EXERCISE 3.3

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 Yes Yes No Yes Yes No Yes Yes Yes 1586 Yes No No No No No No No No 275 No 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

2). Using divisibility tests, determine which of the following numbers are divisible by 4 and 8
a) 572
Ans.
72 is the last two digits. Since 72 is divisible by 4. Hence, 572 is also divisible by 4. 572 is the last three digits. Since 572 is not divisible by 8, it is not divisible by 8.

b) 726352
Ans.  52 is the last two digits and is divisible by 4, therefore the number 726352 is divisible by 4. 352 is the last three digits and is divisible by 8, therefore 726352 is divisible by 8.

c)5500
Ans. 00 are the last 2 digits, therefore 5500 is divisible by 4. 500 is the last 3 digits which is not divisible by 8. Hence 5,500 is not divisible by 8

d) 6000
Ans. 00 is the last 2 digits. Hence 6000 is divisible by 4. 000 is the last 3 digits. Hence 6000 is divisible by 8.

e) 12159
Ans. Since the last 2 digits 59 is not divisible by 4 the 12159 is not divisible by 4, therefore 12159 is not divisible by 4.
Since the last 3 digits 159 are not divisible by 8, therefore 12159 is not divisible by 8.

f) 14560
Ans.  Since the last 2 digits 60 is divisible by 4, 14560 is divisible by 4. Since the last three 3 digits 560 is divisible by 8, 14560 is divisible by 8.

g) 21084
Ans. Since 84 is divisible by 4, 21084 is divisible by 4. Since 084 is not divisible by 8, 21084 is not divisible by 8.

h) 31795072
Ans. Since 72 is divisible by 4, 31795072 is divisible by 4.
Since 072 is divisible by 8, 31795072 is divisible by 8.

i) 1700
Ans. Since the last two digits are 00, 1700 is divisible by 4.
Since 700 is not divisible by 8, 1700 is not divisible by 8.

j) 2150
Ans. Since 50 is not divisible by 4, 2150 is not divisible by 4. Since 150 is not divisible by 8, 2150 is not divisible by 8.

3). Using divisibility tests, determine which of the following numbers are divisible by 6
a) 297144

Ans. Since the last digit 4 is divisible by 2, 297144 is divisible by 2. Adding all the digits of the number we get 27 which is a multiple of 3 , therefore 297144 is divisible by 2 and 3 and hence divisible by 6.

b) 1258
Ans.  Since 8 is divisible by, therefore 1258 is divisible by 2 adding all the digits we get 16 which is not divisible by 3, therefore 1258 is not divisible by 3. Since 1258 is not divisible by 2 and 3 both, the number is not divisible by 6.

b) 4335
Ans. Since 5 is not divisible by 2, 4335 is not divisible by 2. Sum of all digits of the number is 15 which is divisible by 3, therefore 4335 is divisible by 3. Since the number is not divisible by 2 and 3 both, 4335 is not divisible by 6.

c) 61233
Ans.  Since 3 is not divisible by 2, 61233 is not divisible by 2. Sum of all the digit of the number is 15 which is divisible by 3. Hence 61233 is divisible by 3. Since the number is not divisible by 2 and 3 both, 61233 is not divisible by 6.

d) 901352
Ans. Since 2 is divisible by 2, 901352 is divisible by 901352. Sum of all the digits of the number is 20.which is not divisible by 3, therefore 901352 is not divisible by 3. Since the number is not divisible by 2 and 3 both, 901352 is not divisible by 6.

e) 438750
Ans.   Since the last digit is 0, the number is divisible by 2. Sum of all the digits is 27  which is divisible by 3 , therefore the number is divisible by 3. Since the number is divisible by 2 and 3 both, 438750 is divisible by 6.
f) 1790184
Ans. Since 4 is divisible by 2, the number is divisible by 2. Sum of all the digits is 30, which is divisible by 3, therefore the number is divisible by 3. Since the number is divisible by 2 and 3 both, 1790184 is divisible by 6.

g) 12583
Ans.  Since 3 is not divisible by 2, the number is not divisible by 2. Sum of all the digits is 19 which is not divisible by 3, therefore 12583 is not divisible by 3. Since the number id not divisible by 2 and 3, 12583 is not divisible by 6.

g) 639210
Ans.  Since the last digit is 0, the number is divisible by 2. Sum of all the digits is 21 which is divisible by 3, therefore the number is divisible by 3. Since the number is divisible by 2 and 3, 639210 is divisible by 6.

h) 17852
Ans.  Since 2 is divisible by 2, 17852 is divisible by 2. Sum of all the digits is 23 which is not divisible by 3, therefore 17852 is not divisible by 3. Since the number is not divisible by 2 and 3, 17852 is not divisible by 6.

4). Using divisibility tests, determine which of the following numbers are divisible by 11.
a) 5445

Ans.
Sum of the digits at odd places = 5 + 4 = 9
Sum of the digits at even places = 4 + 5 = 9
Difference = 9 – 9 = 0
Since the difference is 0, 5445 is divisible by 11.

b) 10824
Ans
Sum of all the digits at odd places = 4 + 8 + 1 = 13
Sum of all the digits at even places = 2 + 0 = 2
Difference = 13 – 2 = 11
Since the difference is 11 which is divisible by 11, 10824 is divisible by 11.

c) 7138965
Ans.
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
Since the difference 9 is not divisible by 11, 7138965 is not divisible by 11.

d) 70169308
Ans
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
Since the difference is 0, 70169308 is divisible by 11.

e) 10000001
Ans.
Sum of digits at odd places = 1 + 0 + 0 + 0 = 1
Sum of digits at even places = 0 + 0 + 0 + 1 = 1
Difference = 1 – 1 = 0
Since the difference is 0, 10000001 is divisible by 11.

f) 901153
Ans.
Sum of digits at odd places = 3 + 1 + 0 = 4
Sum of digits at even places = 5 + 1 + 9 = 15
Difference 15 – 4 = 11
Since the difference is 11, which is divisible by 11, 901153 is divisible by 11.

5). Write the smallest digit and the largest digit in the blank space of each of the following numbers so that the number is divisible by 3:
a) ……6724
Ans.
Sum of the given digits = 19 (consider nos. from 0 – 9) on adding 2 to 19 we get 2 + 19 = 21 which is divisible by 3. Hence 2 is the smallest number.
Sum of the digits = 19
On adding 9 we get 19 + 9 = 28 which is not divisible by 3.
On adding 8 we get 19 + 8 = 27 which is divisible by 3.
therefore, 8 is the greatest digit.
b) 4765……2
Ans.
Sum of the given digits = 24.
Since 24 is already divisible by 3, the smallest no. that can be replaced is 0.
On adding 9 to the sum we get, 9 + 24 = 33. Which is divisible by 3, therefore 9 is the greatest no.

6). Write a digit in the blank space of each of the following numbers so that the number is divisible by 11:
a) 92…….389
Ans.
let ‘x’ be the no. to be placed here
Sum of digits at odd places = 9 + 3 + 2 = 14
Sum of digits at even places = 8 + x + 9 = 17 + a
Difference = 17 + a – 14 = 3 + a
If the difference is 0 or divisible by 11, then the no. is divisible by 11.
At a = 8, 3 + 8 = 11 which is divisible by 11, therefore the required digit is 8.

b) 8…….9484
Ans.
let ‘x’ be the no. to be placed
Sum of digits at odd places = 4 + 4 + x = 8 + x
Sum of digits at even places = 8 + 9 + 8 = 25
Difference = 25 – (8 + x) = 17 – x
If the difference is 0 or divisible by 11, then the no. is divisible by 11
at a = 6,  17 – 6 = 11 which is divisible by 11 \The required digit is 6.