## 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 │ ? │ ? │ ? │ ? │ ? │ ? │ ? │ ? │ ? │ └───────┴───┴───┴────┴────┴────┴────┴────┴───┴───┘Answer:

┌───────┬────────────────────────────────────────┐ │ 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**

Answer:

(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**

Answer:

(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**

Answer:

(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**

Answer:

(a) __6724

We know that a number is divisible by 3 if the sum of all digits is divisible by 3.

∴ Smallest digit : 2

**2**6724 = 2 + 6 + 7 + 2 + 4 = 21

Largest digit : 8

**8**6724 = 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 4765

**0**2 = 4 + 7 + 6 + 5 + 0 + 2 = 24

Largest digit : 9 4765

**9**2 = 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

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