Divisibility questions are very often asked in most of the Aptitude tests, which can be very easily solved if the simple concept given below is followed. The tabular column drawn below will depict a clear picture of solving the Number quests. Look at them and check the solved problems below. It will definitely add value to your intelligence quotient!
Explanation
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Examples
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Divisibility by 2:
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Even number
Or
Ends with 0,2,4,6 or 8
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23658 ü
Even number
Ends with 8
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4563 û
Odd number
Ends with 3
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Divisibility by 3:
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Sum of its digits is divisible by 3
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23517 ü
2+3+5+1+7= 18
18 is divisible by 3
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45326 û
4+5+3+2+6=20
20 is not divisible by 3
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Divisibility by 4:
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Last two digits is divisible by 4
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1024ü
24 is divisible by 4
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3202 û
02 is not divisible by 4
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Divisibility by 5:
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Ends with 0 or 5
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3045 ü
Ends with 5
1050
Ends with 0
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4523 û
Ends with 3
1032
Ends with 2
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Divisibility by 6:
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Divisible by both 2 and 3
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34314ü
Divisible by both 2 and 3
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3243 û
Divisible by 3 not 2
2048
Divisible by 2 not 3
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Divisibility by 8:
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Last 3 digits is divisible by 8
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17432ü
432 is divisible by 8
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5142 û
142 is not divisible by 8
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Divisibility by 9:
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Sum of its digits is divisible by 9
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43722ü
4+3+7+2+2 = 18
18 is divisible by 9
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3716 û
3+7+1+6 = 17
17 is not divisible by 9
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Divisibility by 10:
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Ends with 0
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2310ü
Ends with 0
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4567 û
Ends with 7
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Divisibility by 11:
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(sum of digits at odd places) – (sum of digits at even places) à 0 or number divisible by 11
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86515ü
(8+5+5)-(6+1)=11
11 is divisible by 11
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234764 û
(3+7+4) – (2+4+6) = 2
2 is not divisible by 11
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Types of numbers:
1. Natural numbers -1, 2, 3, …
2. Whole numbers – 0, 1, 2, 3, …
3. Integers - …, -3, -2, -1, 0, 1, 2, 3, …
a. Positive integers – 1, 2, 3, …
b. Negative integers – -1, -2, -3, …
c. Non negative integers – 0, 1, 2, 3, …
d. Non positive integers – 0, -1, -2, -3, …
4. Even numbers – 2, 4, 6, 8, …
5. Odd numbers – 1, 3, 5, …
6. Prime numbers – 2, 3, 5, 7, 11, …
Prime numbers up to 100 :
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 , 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97
7. Composite numbers – 4, 6, 8, 9, 10, 12, ...
Aptitude questions:
1. Which one of the following numbers is exactly divisible by 11? [ C.D.S 2003]
(a) 235641 (b) 245642 (c) 315624 (d) 415624
Ans: d
Solution:
(3+6+1) -(2+5+4) = 10-11 = -1 à not divisible by 11
(4+6+2) – (2+5+4) = 12-11 =1 à not divisible by 11
(1+6+4) – (3+5+2) = 11-10 =1 à not divisible by 11
(1+6+4) – (4+5+2) = 11-11 =0 à divisible by 11
2. If 1*548 is divisible by 3, which of the following digits can replace *? [S.S.C. 1999]
(a) 0 (b) 2 (c) 7 (d) 8
Ans: a
Solution:
If a number is divisible by 3 then sum of its digits is divisible by 3
So,
Sum of digits = 1+*+5+4+8 = 18+* à Only if * = 0, the sum will be divisible by 3
3. 6897 is divisibe by : [I.A.M. 2002]
(a) 11 only (b) 19 only (c) both 19 and 11 (d) neither 19 nor 11
Ans: (c)
Solution:
(6+9)-(8+7) = 15-15 = 0 à Divisible by 11
6897/19 = 363 à Divisible by 19
4. The sum of three consecutive odd numbers is always divisible by :
(i) 2 (ii) 3 (iii) 5 (iv) 6 [Hotel Mgmt, 2003]
(a) Only (i) (b) Only (ii) (c) Only (i) and (iii) (d) Only (ii) and (iv)
Ans: (b)
Solution:
Let 3 consecutive odd numbers be 2x+1, 2x+3, 2x+5,
Sum = 6x+9 = 3(2x+3) à divisible of 3
5. The difference between the squares of two consecutive odd integers is always divisible by : [M.B.A. 2003]
(a) 3 (b) 6 (c) 7 (d) 8
Ans: d
Solution:
Let 2 consecutive odd numbers be 2x+1, 2x+3
Difference of squares = (2x+3)2-(2x+1)2 = (4x2+9+12x)-(4x2+1+4x)
=4 x2+9+12x-4x2-1-4x = 8+8x=8(x+1)
=4 x2+9+12x-4x2-1-4x = 8+8x=8(x+1)
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