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.
