**Class 6 | Mathematics | Chapter 3| Playing with Numbers | Exercise 3.3**

**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):

Number | |||||||||

2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 | |

128 | |||||||||

990 | |||||||||

1586 | |||||||||

1586 | |||||||||

275 | |||||||||

6686 | |||||||||

639210 | |||||||||

429714 | |||||||||

2856 | |||||||||

3060 | |||||||||

406839 |

**Answer 1**

Number | |||||||||

2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 | |

128 | Yes | No | Yes | No | No | Yes | No | No | No |

990 | Yes | Yes | No | No | Yes | No | Yes | Yes | Yes |

1586 | Yes | No | No | No | No | No | No | No | No |

1586 | No | No | No | No | No | No | No | No | No |

275 | Yes | No | No | No | No | No | No | No | Yes |

6686 | Yes | No | No | No | No | No | No | No | No |

639210 | Yes | Yes | No | No | 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 | No | Yes | No | Yes | Yes | No |

406839 | No | Yes | No | No | No | No | No | No | No |

**Question 2:** Using divisibility test, determine which of the following numbers are divisibly by 4; by 8:

(a) 572 (b) 726352 (c) 5500 (d) 6000 (e) 12159 (f) 14560 (g) 21084 (h) 31795072 (i) 1700 (j) 2150

**Answer 2:**

**(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.

**Question 3:** 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 3:**

**(a)** 297144 : Divisible by 2 as its units place is an even number.

Divisible by 3 as sum of its digits (= 27) is divisible by 3.

Since the number is divisible by both 2 and 3, therefore, it is also divisible by 6.

**(b)** 1258 : Divisible by 2 as its units place is an even number.

Not divisible by 3 as sum of its digits (= 16) is not divisible by 3.

Since the number is not divisible by both 2 and 3, therefore, 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, therefore, it is not divisible by 6.

**(d)** 61233 : 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, therefore, it is not divisible by 6.

**(e)** 901352 : Divisible by 2 as its units place is an even number.

Not divisible by 3 as sum of its digits (= 20) is not divisible by 3.

Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.

**(f)** 438750 : Divisible by 2 as its units place is an even number.

Divisible by 3 as sum of its digits (= 27) is not divisible by 3.

Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.

**(g)** 1790184 : Divisible by 2 as its units place is an even number.

Divisible by 3 as sum of its digits (= 30) is not divisible by 3.

Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.

**(h)** 12583 : Not divisible by 2 as its units place is not an even number.

Not divisible by 3 as sum of its digits (= 19) is not divisible by 3.

Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.

**(i)** 639210 : Divisible by 2 as its units place is an even number.

Divisible by 3 as sum of its digits (= 21) is not divisible by 3.

Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.

**(j)** 17852 : Divisible by 2 as its units place is an even number.

Not divisible by 3 as sum of its digits (= 23) is not divisible by 3.

Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.

**Question 4:** 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 4:** **(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 + 3 + 7 = 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.

**Question 5:** Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisibly by 3:

(a) __6724

(b) 4765 __ 2

**Answer 5: **(a) We know that a number is divisible by 3 if the sum of all digits is divisible by 3.

Therefore,

Smallest digit : 2 26724 = 2 + 6 + 7 + 2 + 4 = 21

Largest digit : 8 86724 = 8 + 6 + 7 + 2 + 4 = 27

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

Therefore, Smallest digit : 0 476502 = 4 + 7 + 6 + 5 + 0 + 2 = 24

Largest digit : 9 476592 = 4 + 7 + 6 + 5 + 0 + 2 = 33

**Question 6:** Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisibly by 11:

(a) 92 __389

(b) 8 ___ 9484

**Answer 6:** (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