**Playing with Numbers**

**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 | |||||||||

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

128 | |||||||||

990 | |||||||||

1586 | |||||||||

275 | |||||||||

6686 | |||||||||

639210 | |||||||||

429714 | |||||||||

2856 | |||||||||

3060 | |||||||||

406839 | |||||||||

Number | |||||||||

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 | No |

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

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

429714 | Yes | Yes | No | No | Yes | No | Yes | No | No |

2856 | Yes | Yes | No | No | Yes | No | 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; 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

The number formed by the last two digits (digit in Unit’s place and digit in Ten’s place = 72

72 is divisible by 4

Therefore, 572 is divisible by 4

The number formed by the last three digits is 572

572 is not divisible by 8

(b) 726352

The number formed by the last two digits in 726352 is 52

52 is divisible by 4

Therefore, 726352 is divisible by 4

The number formed by the last three digits is 352

352 is divisible by 8

Therefore, 726352 is divisible by 8

(c) 5500

The number formed by the last two digits in 5500 is 00

00 is divisible by 4

Therefore, 5500 is divisible by 4

The number formed by the last three digits is 500

500 is not divisible by 8

Therefore, 5500 is not divisible by 8

(d) 6000

The number formed by the last two digits in 6000 is 00

00 is divisible by 4

Therefore, 6000 is divisible by 4

The number formed by the last three digits is 000

000 is divisible by 8

Therefore, 6000 is divisible by 8

(e) 12159

The number formed by the last two digits in 12159 is 59

59 is not divisible by 4

Therefore, 12159 is not divisible by 4

The number formed by the last three digits is 159

159 is not divisible by 8

Therefore, 12159 is not divisible by 8

(f) 14560

The number formed by the last two digits in 14560 is 60

60 is divisible by 4

Therefore, 14560 is divisible by 4

The number formed by the last three digits is 560

560 is divisible by 8

Therefore, 14560 is divisible by 8

(g) 21084

The number formed by the last two digits in 21084 is 84

84 is divisible by 4

Therefore, 21084 is divisible by 4

The number formed by the last three digits is 084

084 is not divisible by 8

Therefore, 21084 is not divisible by 8

(h) 31795072

The number formed by the last two digits in 31795072 is 72

72 is divisible by 4

Therefore, 31795072 is divisible by 4

The number formed by the last three digits is 072

072 is divisible by 8

Therefore, 31795072 is divisible by 8

(i) 1700

The number formed by the last two digits in 1700is 00

00 is divisible by 4

Therefore, 1700 is divisible by 4

The number formed by the last three digits is 700

700 is not divisible by 8

Therefore, 1700 is not divisible by 8

(j) 2150

The number formed by the last two digits in 2150 is 50

50 is not divisible by 4

Therefore, 2150 is not divisible by 4

The number formed by the last three digits is 150

150 is not divisible by 8

Therefore, 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

Answer:

(a) 297144

297144 is an even number, therefore it is divisible by 2.

The sum of the digits 2+9+7+1+4+4 = 27 is divisible by 3.

Therefore, 297144 is divisible by 6.

(b) 1258

1258 is an even number, therefore it is divisible by 2.

The sum of the digits 1+2+5+8 = 16 is not divisible by 3.

The given number 1258 is divisible by 2 but not by 3.

Therefore, 1258 is not divisible by 6.

(c) 4335

4335 is an odd number, therefore it is not divisible by 2.

The sumof the digits 4+3+3+5 = 15 is divisible by 3.

The given number 4335 is not divisible by 2 but divisible by 3.

Therefore, 4335 is not divisible by 6.

(d) 61233

61233 is an odd number, therefore it is not divisible by 2.

The sum of the digits 6+1+2+3+3 = 15 is divisible by 3.

The given number 61233 is not divisible by 2 but divisible by 3.

Therefore, 61233 s not divisible by 6.

(e) 901352

901352 is an even number, therefore it is divisible by 2.

The sum of the digits 9+0+1+3+5+2 = 20 is not divisible by 3.

The given number 901352 is divisible by 2 but not by 3.

Therefore, 901352 is not divisible by 6.

(f) 438750

438750 is an even number, therefore it is divisible by 2.

The sum of the digits 4+3+8+7+5+0 = 27 is divisible by 3.

The given number 438750 is divisible by 2 and 3.

Therefore, 438750 is divisible by 6.

(g) 1790184

438750 is an even number, therefore it is divisible by 2.

The sum of the digits 4+3+8+7+5+0 = 27 is divisible by 3.

The given number 438750 is divisible by 2 and 3.

Therefore, 438750 is divisible by 6.

(h) 12583

12583 is an odd number, therefore it is not divisible by 2.

Sum of the digits 1+2+5+8+3 = 19 is not divisible by 3.

The given number 12583 is not divisible by 2 and 3.

Therefore, 12583 is not divisible by 6.

(i) 639210

639210 is an even number, therefore it is divisible by 2.

The sum of the digits 6+3+9+2+1+0= 21 is divisible by 3.

The given number 639210is divisible by 2 and 3.

Therefore, 639210 is divisible by 6.

(j) 17852

17852is an even number, therefore it is divisible by 2.

The sum of the digits 1+7+8+5+2 = 23 is not divisible by 3.

The given number 17852 is divisible by 2 but not by 3.

Therefore, 17852 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

Answer:

(a) 5445

Sum of the digits in odd place is 5 + 4 = 9

Sum of the digits in even place is 4 + 5 = 9

Subtraction of 9-9 = 0

It is multiple of 11

Therefore, 5445 is divisible by 11.

(b) 10824

Sum of the digits in odd place is 4+8+1 = 13

Sum of the digits in even place is 2+0 = 2

Subtraction of 13-2 = 11

It is multiple of 11

Therefore, 10824 is divisible by 11.

(c) 7138965

Sum of the digits in odd place is 5+9+3+7 = 24

Sum of the digits in even place is 6+8+1 = 15

Subtraction of 24-15 = 9

It is not a multiple of 11

Therefore, 7138965 is not divisible by 11

(d) 70169308

Sum of the digits in odd place is 8+3+6+0 = 17

Sum of the digits in even place is 0+9+1+7 = 17

Subtraction of 17-17 = 0

It is a multiple of 11

Therefore, 70169308 is divisible by 11

(e) 10000001

Sum of the digits in odd place is 1+0+0+0 = 1

Sum of the digits in even place is 0+0+0+1 = 1

Subtraction of 1-1 = 0

It is a multiple of 11

Therefore, 10000001 is divisible by 11

(f) 901153

Sum of the digits in odd place is 3+1+0 = 4

Sum of the digits in even place is 5+1+9 = 15

Subtraction of 15-1 = 11

It is a multiple of 11

Therefore, 901153 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

Smallest number – 2

Greatest number – 8

The sum

A number is divisible by 3 if the sum of the digits in the number is a multiple of 3.

The sum of the digits is 6+7+2+4 = 19

If we add 2 then the sum will be 21 and it is a multiple of 3.

If we add 8 then the sum will be 27 and it is a multiple of 3.

(b) 4765 __ 2

Smallest number – 0

Greatest number – 9

Explanation:

A number is divisible by 3 if the sum of the digits in the number is a multiple of 3.

The sum of the digits is 4+7+6+5+2 = 24

24 itself is a multiple of 3 therefore smallest number to add is 0.

If we add 0 then the sum will be 24 and it is a multiple of 3.

If we add 9 then the sum will be 33 and it is a multiple of 3.

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

Number in the blank – 8

Explanation:

A number is divisible by 11 if the sum of the digits in the odd place and The sum of the digits in the even place is either 0 or a multiple of 11.

The sum of the digits in odd place is 9+3+2 = 14

Sum of the digits in even place is 8+9 = 17 (excluding the blank)

Subtraction of 17-14 = 3

The number to be added is in the even place, so that subtraction is 11 then the given number is divisible by 11.

Instead of 17 if the total is 25 then the given number will be divisible by 11.

25-17 = 8.

If we add 8 then the given number will be divisible by 11.

(b) 8 __ 9484

Number in the blank – 6

Explanation:

A number is divisible by 11 if the sum of the digits in the odd place and the sum of the digits in the even place is either 0 or a multiple of 11.

Sum of the digits in odd place is 4+4 = 8 (excluding the blank)

Sum of the digits in even place is 8+9+8 = 25

Subtraction of 25-8 = 17

The number to be added is in the odd place so that subtraction is 11 then the given number is divisible by 11.

Instead of 8 if the total is 14 then the given number will be divisible by 11.

14-8 = 6.

If we add 6 then the given number will be divisible by 11.