### Chapter - 3 Playing with Numbers

Q
##### Playing With Numbers

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): Solution: 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) 31795072   (i) 1700   (j) 2150

Solution:

(a) 572 → 72 are the last two digits of 572. 72 is divisible by 4. Hence, 572 is also divisible by 4.

572 are the last three digits of 572. 572 is not divisible by 8. Hence, 572 is not divisible by 8.

(b) 726352 → 52 are the last two digits of 726352. 52 is divisible by 4. Hence, 726352 is divisible by 4.

352 are the last three digits of 726352. 352 is divisible by 8. Hence, 726352 is divisible by 8.

(c) 5500 → Last two digits of 5500 are 00. Hence 5500 is divisible by 4.

500 are the last three digits of 5000. 500 is not divisible by 8. Hence, 5500 is not divisible by 8.

(d) 6000 → Last two digits of 6000 are 00. Hence 6000 is divisible by 4.

Last three digits of 600 are 000. Hence, 6000 is divisible by 8.

(e) 12159 → 59 are the last two digits of 12159. 59 is not divisible by 4. Hence,12159 is not divisible by 4.

159 are the last three digits. 159 is not divisible by 8. Hence, 12159 is not divisible by 8.

(f) 14560 → 60 are the last two digits of 14560. 60 is divisible by 4. Hence, 14560 is divisible by 4.

560 are the last three digits. 560 is divisible by 8. Hence, 14560 is divisible by 8.

(g) 21084 → 84 are the last two digits of 21084. 84 is divisible by 4. Hence, 21084 is divisible by 4.

084 are the last three digits. 084 is not divisible by 8. Hence, 21084 is not divisible by 8.

(h) 3179507 → 72 are the last two digits. 72 is divisible by 4. Hence, 31795072 is divisible by 4.

072 are the last three digits. 072 is divisible by 8. Hence, 31795072 is divisible by 8.

(i) 1700 → Last two digits of 1700 are 00. Hence, 1700 is divisible by 4.

700 are the last three digits. 700 is not divisible by 8. Hence, 1700 is not divisible by 8.

(j) 2150 → 50 are the last two digits of 2150. 50 is not divisible by 4. Hence, 2150 is not divisible by 4.

150 are the last three digits. 150 is not divisible by 8. Hence, 2150 is not 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: (a) 297144

The last digit of the number is 4. Hence, the number is divisible by 2.

By adding all the digits of the number, we get 27 which is divisible by 3. Hence, the number is divisible by 3.

The number is divisible by both 2 and 3. Therefore, the number is divisible by 6.

(b) 1258

The last digit of the number is 8. Hence, the number is divisible by 2.

By adding all the digits of the number, we get 16 which is not divisible by 3. Hence, the number is not divisible by 3.

The number is not divisible by both 2 and 3. Therefore, the number is not divisible by 6.

(c) 4335

The last digit of the number is 5 which is not divisible by 2. Hence, the number is not divisible by 2.

By adding all the digits of the number, we get 15 which is divisible by 3. Hence, the number is divisible by 3.

The number is not divisible by both 2 and 3. Therefore, the number is not divisible by 6.

(d) 61233

The last digit of the number is 3 which is not divisible by 2. Hence, the number is not divisible by 2.

By adding all the digits of the number, we get 15 which is divisible by 3. Hence, the number is divisible by 3.

The number is not divisible by both 2 and 3. Therefore, the number is not divisible by 6.

(e) 901352

The last digit of the number is 2. Hence, the number is divisible by 2.

By adding all the digits of the number, we get 20 which is not divisible by 3. Hence, the number is not divisible by 3.

The number is not divisible by both 2 and 3. Therefore, the number is not divisible by 6.

(f) 438750

The last digit of the number is 0. Hence, the number is divisible by 2.

By adding all the digits of the number, we get 27 which is divisible by 3. Hence, the number is divisible by 3.

The number is divisible by both 2 and 3. Therefore, the number is divisible by 6.

(g) 1790184

The last digit of the number is 4. Hence, the number is divisible by 2.

By adding all the digits of the number, we get 30 which is divisible by 3. Hence, the number is divisible by 3.

The number is divisible by both 2 and 3. Therefore, the number is divisible by 6.

(h) 12583

The last digit of the number is 3. Hence, the number is not divisible by 2.

By adding all the digits of the number, we get 19 which is not divisible by 3. Hence, the number is not divisible by 3.

The number is not divisible by both 2 and 3. Therefore, the number is not divisible by 6.

(i) 639210

The last digit of the number is 0. Hence, the number is divisible by 2.

By adding all the digits of the number, we get 21 which is divisible by 3. Hence, the number is divisible by 3.

The number is divisible by both 2 and 3. Therefore, the number is divisible by 6.

(j) 17852

The last digit of the number is 2. Hence, the number is divisible by 2.

By adding all the digits of the number, we get 23 which is not divisible by 3. Hence, the number is not divisible by 3.

The number is not divisible by both 2 and 3. Hence, the number 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: (a) 5445

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

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

Difference of both sums = 9 - 9 = 0

Since the difference is 0, therefore, the number is divisible 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 of both sums = 13 - 2 = 11

Since the difference is 11, therefore, the number is divisible by 11.

(c) 7138965

Sum of the digits at odd places = 5 + 9 + 347 = 24

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

Difference of both sums = 24 -15 = 9

Since the difference is neither 0 nor 11, therefore, the number is not divisible by 11.

(d) 70169308

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

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

Difference of both sums = 17 - 17 = 0

Since the difference is 0, therefore, the number is divisible by 11.

(e) 10000001

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

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

Difference of both sums = 1 - 1 = 0

Since the difference is 0, therefore, the number 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 of both sums = 15 - 4 = 11

Since the difference is 11, therefore, the number 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: (a) __6724

Sum of the given digits = 19

The Sum of its digit should be divisible by 3 to make the number divisible by 3.

Since, 21 is the smallest multiple of 3 which comes after 19.

So, smallest number = 21 - 19 = 2

Now, if we put 8, the sum of digits will be 27 which is divisible by 3.

Therefore the number will be divisible by 3.

Hence, the largest digit is 8.

(b) 4765__2

Sum of the given digits = 24

The Sum of its digits should be divisible by 3 to make the number divisible by 3.

Since 24 is already divisible by 3.

Hence, the smallest number that can be replaced is 0.

Now, If we put 9, the sum of its digits becomes 33 Since. 33 is divisible by 3.

Therefore the number will be divisible by 3.

Hence, the largest digit 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: (a) 92 __ 389

Let the digit be 'x'.

We know, the difference of the sum of the digits at odd places and that of even places should be either 0 or a multiple of 11, then the number is divisible by 11.

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

Sum of its digits at even places = 8 + x + 9 = 17 + x

Difference of both sums  = 17 + x - 14 = 3 + x

The difference should be 0 or a multiple of 11, then the number is divisible by 11.

If 3 + x = 0

x = - 3

But it cannot be negative.

So, let's consider another case.

If 3 + x = 11

x = 11 - 3 = 8

Therefore the required digit is 8.

(b) 8 __ 9484

Let the digit be 'x'.

We know, the difference of the sum of the digits at odd places and that of even places should be either 0 or a multiple of 11, then the number is divisible by 11.

Sum of its digits at odd places = x + 4 + 4 = 8 + x

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

Difference of both sums = 25 - (8 + x) = 17 - x

The difference should be 0 or a multiple of 11, then the number is divisible by 11

If 17 - x = 0

x = 17

We required a digit, not a number so, x = 17 is not possible.

Now, let's consider another case.

If 17 - x = 11

x = 17 - 11 = 6

Therefore the required digit is 6. 