## Playing With Numbers

EXERCISE 3.3
CBSE Class 6 Maths

Q1: 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   │ Y │ N │ Y  │ N  │ N  │ Y  │ N  │ N │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 990   │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 1586  │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 275   │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 6686  │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│639210 │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│429714 │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 2856  │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 3060  │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│406839 │ ? │ ? │ ?  │ ?  │ ?  │ ?  │ ?  │ ? │ ? │
└───────┴───┴───┴────┴────┴────┴────┴────┴───┴───┘

```

```
┌───────┬────────────────────────────────────────┐
│ Number│          Divisible By                  │
├───────┼───┬───┬────┬────┬────┬────┬────┬───┬───┤
│       │ 2 │ 3 │ 4  │ 5  │ 6  │ 8  │ 9  │ 10│11 │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 128   │ Y │ N │ Y  │ N  │ N  │ Y  │ N  │ N │ N │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 990   │ ✔ │ ✔ │ ✗  │ ✔  │ ✔  │ ✗  │ ✔  │ ✔ │ ✔ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 1586  │ ✔ │ ✗ │ ✗  │ ✗  │ ✗  │ ✗  │ ✗  │ ✗ │ ✗ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 275   │ ✗ │ ✗ │ ✗  │ ✔  │ ✗  │ ✗  │ ✗  │ ✗ │ ✔ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 6686  │ ✔ │ ✗ │ ✗  │ ✗  │ ✗  │ ✗  │ ✗  │ ✗ │ ✗ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│639210 │ ✔ │ ✔ │ ✗  │ ✔  │ ✔  │ ✗  │ ✗  │ ✔ │ ✔ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│429714 │ ✔ │ ✔ │ ✗  │ ✗  │ ✔  │ ✗  │ ✔  │ ✗ │ ✗ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 2856  │ ✔ │ ✔ │ ✔  │ ✗  │ ✔  │ ✔  │ ✗  │ ✗ │ ✗ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│ 3060  │ ✔ │ ✔ │ ✔  │ ✔  │ ✔  │ ✗  │ ✔  │ ✔ │ ✗ │
├───────┼───┼───┼────┼────┼────┼────┼────┼───┼───┤
│406839 │ ✗ │ ✔ │ ✗  │ ✗  │ ✗  │ ✗  │ ✗  │ ✗ │ ✗ │
└───────┴───┴───┴────┴────┴────┴────┴────┴───┴───┘

```

Q2: Using divisibility test, determine which of the following numbers are divisible by 4 and by 8:
(a) 572
(b) 726352
(c) 5500
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150

(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) 31795072
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.

(i) 1700
Divisible by 4 as its last two digits are 0.
Not divisible by 8 as its last three digits are not divisible by 8.

(j) 5500
Not divisible by 4 as its last two digits are not divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.

Q3: Using divisibility test, 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

(a) 297144
Divisible by 2 as its units place is an even number.
Divisible by 3 as sum of its digits (= 27).
Since the number is divisible by both 2 and 3, ∴ it is also divisible by 6.

(b) 1258
Divisible by 2 as its units place is an even number.
Sum of its digits (1+2+5+8 = 16) is not divisible by 3.
Since the number is not divisible by both 2 and 3, ∴ it is not divisible by 6.

(c) 4335
Not divisible by 2 as its units place is not an even number.
Divisible by 3 as sum of its digits (= 15) is divisible by 3.
Since the number is not divisible by both 2 and 3, ∴ it is not divisible by 6.

(d) 61233
Since it is not an even number, it is not divisible by 2.
Sum of its digits (= 15), it is divisible by 3.
Since the number is not divisible by both 2 and 3, ∴ it is not divisible by 6.

(e) 901352
It is an even number, it is divisible by 2.
As sum of its digits (= 20), it is not divisible by 3.
Since the number is not divisible by both 2 and 3, ∴ it is not divisible by 6.

(f) 438750
An even number, it is divisible by 2
Sum of its digits (= 27), it is not divisible by 3.
Since the number is divisible by both 2 and 3, ∴ it is divisible by 6.

(g) 1790184
Divisible by 2 as it is an even number.
As sum of its digits (= 30) is divisible by 3.
Since the number is divisible by both 2 and 3, ∴ it is divisible by 6.

(h) 12583
Divisible by 2 as its units place is an even number.
Not Divisible by 3 as sum of its digits (= 19 not a multiple of 3).
Since the number is not divisible by both 2 and 3, ∴ it is not divisible by 6.

(i) 639210
Divisible by 2 as it is an even number.
Divisible by 3 as sum of its digits (= 21).
Since the number is divisible by both 2 and 3, ∴ it is divisible by 6.

(j) 17852
Divisible by 2 as it is an even number.
Not divisible by 3 as sum of its digits (= 23) .
Since the number is not divisible by both 2 and 3, ∴ it is not divisible by 6.

Q4: Using divisibility test, determine which of the following numbers are divisible by 11:
(a) 5445
(b) 10824
(c) 7138965
(d) 70169308
(e) 10000001
(f) 901153

(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
∵ the difference is 0, ∴ 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
∵ the difference is 11, ∴ the number is divisible 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 of both sums = 24 – 15 = 9
∵ the difference is neither 0 nor 11, ∴ the number is 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
∵ the difference is 0, ∴ 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
∵ the difference is 0, ∴ 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
∵ the difference is 11, ∴ the number is divisible by 11.

Q5: Write the smallest digit and the largest digit in the blanks space of each of the following
numbers so that the number formed is divisible by 3:

(a) __6724

(b) 4765__2

(a) __6724
We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
∴ Smallest digit : 2 26724 = 2 + 6 + 7 + 2 + 4 = 21
Largest digit : 8 86724 = 8 + 6 + 7 + 2 + 4 = 27

(b) 4765__2
We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
∴ Smallest digit : 0 476502 = 4 + 7 + 6 + 5 + 0 + 2 = 24
Largest digit : 9 476592 = 4 + 7 + 6 + 5 + 0 + 2 = 33

Q6: Write the smallest digit and the largest digit in the blanks space of each of the following
numbers so that the number formed is divisible by 11:

(a) 92_389

(b) 8_9484

Answer: (a) We know that a number is divisible by 11 if the difference of the sum of the digits
at odd places and that of even places should be either 0 or 11.
Therefore, 928389
Odd places = 9 + 8 + 8 = 25
Even places    = 2 + 3 + 9 = 14
difference = 25 – 14 = 11

(b) We know that a number is divisible by 11 if the difference of the sum of the digits
at odd places and that of even places should be either 0 or 11.
Therefore, 869484
Odd places = 8 + 9 + 8 = 25
Even places = 6 + 4 + 4 = 14
Difference = 25 – 14 = 11  