In many competitive questions, it becomes very important to find whether given number is divisible by another given number or not. It is required when we solve HCF (Highest Common Factor), LCM (Least Common Multiple), Composite numbers, Co-Prime numbers, multiples, factors, etc.
So this topic is extensively important in understanding many parts of number mathematics and algebra as well because all operations in number mathematics are always true for algebra and other mathematics also.
Divisibility Rule (Divisibility Test - 1 to 10)
Divisibility test means testing of a number whether it is divisible completely by another given number. It is always means that the first number (Dividend) and the second number (Divisor) are integers. Result of a division is called Quotient and if something is left, it is called a remainder.
For Example:
456382 ÷ 12 = (Q=38031 and R=10)
Important:
"If in a division, quotient is an integer, and remainder is zero, it means that dividend is completely divisible by divisor."
"If in a division, either quotient is a fraction with remainder zero or quotient is integer with remainder non-zero, it means that dividend is not completely divisible by divisor."
1. Divisible by 1: Every integer is divisibility by 1 always. so it is not needed to test any integer to be divisible by 1.
2. Divisible By 2: An integer having 0, 2, 4, 6, 8 at its unity place or Last digit, is divisible by 2.
| Steps of Test: There is only one step of test i.e. check the last digit is 0,2,4,6,8 or number is even number |
For Example:
2340, 12, 34, 56, 35798, etc
Note: The integers which are completely divisible by 2 are called even numbers.
3. Divisible By 3: If sum of all digits in an integer is divisible by 3, then the number is divisible by 3.
| Step for Test: 1. Sum all the digits 2. If sum found is more than one digit number, then again sum all the digits of the previous sum. 3. Repeat the process until we get sum of digits as a one digit number. 4. If final sum is divisible by 3, then the given number is divisible by 3. If final sum is not divisible by 3, then the given number is not divisible by 3. |
For Example:
Example 1: Let us check whether 162738951 is divisible by 3.
Sum of All digits = 1 + 6 + 2 + 7 + 3 + 8 + 9 + 5 + 1
= 42
Again Sum all digits = 4 + 2 = 6 (Divisible by 3)
We can see that the sum of all digits in the given number is divisible by 3, so the given number 162738951 is divisible by 3.
Verification: Now divide the number by 3 using conventional division method, we get that the given number 162738951 is divisible by 3 and quotient is 54246317 with remainder 0.
Example 2: Now let us take another example: Check whether 7635428743 is divisible by 3?
Sum of all digits = 7 + 6 + 3 + 5 + 4 + 2 + 8 + 7 + 4 + 3
= 49
Again sum all digits = 4 + 9 = 13 (Not divisible by 3)
We can see that sum of all digits in the given number is not divisible by 3, so the given number 7635428743 is not divisible by 3.
Verification: Now divide the number by 4 using conventional division method, we get that the given number 7635428743 is not divisible by 3 and quotient is 2545142914 with remainder 1 (≠ 0).
4. Divisible By 4: If combination of Last two place (tens place and unit place digits) forms a number divisible by 4, then whole number is divisible by 4.
| Steps for Test: 1. First check whether last place of the number is 0,2,4,6,8 means number is even. 2. If number is even, then write last two digits (10's place and 1's place) as a number from the given number. 2. If this number is divisible by 4, then the given number is divisible by 4. If this number is not divisible by 4, then the given number is not divisible by 4. |
For Example:
Let us check whether 8456733932 is divisible by 4.
Combination of last two digits in 8456733932 is 32.
If we divide 32, we get that 32 is divisible by 4. So the given number 8456733932 is divisible by 4.
Verification: Now divide the number by 4 using conventional division method, we get that the given number 8456733932 is divisible by 4 and quotient is 2114183483 with remainder 0.
Now let us take another example: Check whether 7635428714 is divisible by 4?
Combination of last two digits in 7635428714 is 14.
If we divide 14, we get that 14 is not divisible by 4. So the given number 7635428714 is not divisible by 4.
Verification: Now divide the number by 4 using conventional division method, we get that the given number 7635428714 is not divisible by 4 and quotient is 1908857178 with remainder 2 (≠ 0).
5. Divisible By 5: If Unit place (Last place) of number has either 0 or 5 then number is divisible by 5.
| Steps of Test: There is only one step of test i.e. check the last digit is 0, 5. |
For Example: 7520, 23456785, 975315, etc.
6. Divisible By 6: If a number is divisible by 2 as well as divisible by 3, then the number is completely divisible by 6.
So it is very clear that the number should follow both the rules of divisibility test by 2 and divisibility test by 3 simultaneously.
Note: If number is not divisible by any one of 2 and 3, then number will not be divisible by 6. If number is divisible by 2 by not divisible by 3, it is not divisible by 6. Similarly, if number is not divisible by 2 but divisible by 3, it is not divisible by 6.
| Steps for Test: 1. Test for divisibility by 2. If last digit of number is 0,2,4,6,8 or number is even then it is divisible by 2. Then we proceed for next step If last digit of number is neither of 0,2,4,6,8 or number is odd, then it is not divisible by 2. Then stop testing and declare that number is not divisible by 6. 2. If Number is divisible by 2, then we proceed for step 3. 3. Test for Divisibility by 3 Sum all the digits If sum found is more than one digit number, then again sum all the digits of the previous sum. Repeat the process until we get sum of digits as a one digit number. If final sum is divisible by 3, then the given number is divisible by 3. If final sum is not divisible by 3, then the given number is not divisible by 3. 4. If number is divisible by 2 and 3, The given number is divisible by 6. |
For Example:
Example 1: Let us check whether 74558232 is divisible by 6.
Step 1: Divisibility Test By 2:
Since last digit of number is 2 which is divisible by 2, so number is divisible by 2.
Step 2: Divisibility Test By 3:
Sum of All digits = 7 + 4 + 5 + 5 + 8 + 2 + 3 + 2 = 36
Again Sum of all digits = 3 + 6 = 9 (Divisible by 3)
So the number is divisible by 3.
Since the given number is divisible by 2 and 3, Hence the given number 74558232 is divisible by 6.
Verification: Now divide the number by 6 using conventional division method, we get that the given number 74558232 is divisible by 6 and quotient is 12426372 with remainder 0
Example 2:Let us take another example: Check whether 32644238 is divisible by 6?
Step 1: Divisibility Test By 2:
Since last digit of number is 8 which is divisible by 2, so number is divisible by 2.
Step 2: Divisibility Test By 3:
Sum of All digits = 3 + 2 + 6 + 4 + 4 + 2 + 3 + 8 = 32
Again Sum of all digits = 3 + 2 = 5 (Not Divisible by 3)
So the number is not divisible by 3.
Since the given number is divisible by 2 but not divisible by 3, Hence the given number 32644238 is not divisible by 6.
Verification: Now divide the number by 6 using conventional division method, we get that the given number 32644238 is not divisible by 6 and quotient is 5440706 with remainder 1 (≠ 0).
Example 3: Let us take another example: Check whether 4876223 is divisible by 6?
Step 1: Divisibility Test By 2:
Since last digit of number is 3 which is not divisible by 2, so number is not divisible by 2.
Since the given number is not divisible by 2, Hence the given number 4876223 is not divisible by 6.
Verification: Now divide the number by 6 using conventional division method, we get that the given number 4876223 is not divisible by 6 and quotient is 812703 with remainder 5 (≠ 0).
7. Divisible By 7: A number is divisible by 7 if result obtained from the subtraction of double of last digit (one's place) from the remaining part of the number is divisible by 7.
| Steps for Test: 1. Write the last digit of number i.e. one's place of the number. 2. Double it (Means multiply it by 2) 3. Subtract this number from the remaining portion of the number. We will obtain a result. 4. Repeat above three steps for the result obtained many times until we get a number less than 70. 5. If final result is divisible by 7, then the given number is divisible by 7. |
For Example:
Example 1: Let us check whether 98994 is divisible by 7.
The unit place of this number is 4
Doubling 4, we get 4 × 2 = 8
Remaining Portion of number = 9899
Subtract double number 9899 - 8 = 9891
Repeat the process: The unit place of new result is 1
Doubling 1, we get 2
Subtract it from new remaining portion 989 - 2 = 987
Repeat the process: The unit place of new result is 7
Doubling 7, we get 14
Subtract it from new remaining portion 98 - 14 = 84
Repeat the process: The unit place of new result is 4
Doubling 4, we get 8
Subtract it from new remaining portion 8 - 4 = 0 (Divisible By 7)
So the given number 98994 is divisible by 7.
Verification: Now divide the number by 7 using conventional division method, we get that the given number 98994 is divisible by 7 and quotient is 14142 with remainder 0.
Example 2: Let us check whether 2268 is divisible by 7.
The unit place of this number is 8
Doubling 8, we get 8 × 2 = 16
Remaining Portion of number = 226
Subtract double number 226 - 16 = 210
Repeat the process: The unit place of new result is 0
Doubling 0, we get 0
Subtract it from new remaining portion 21 - 0 = 21 (Divisible By 7)
So the given number 2268 is divisible by 7.
Verification: Now divide the number by 7 using conventional division method, we get that the given number 2268 is divisible by 7 and quotient is 324 with remainder 0.
8. Divisible By 8: If combination of last three digits (Hundred's, Ten's and one's Place) as a number is divisible by 8, then the given number is divisible by 8.
| Steps for Test: 1. First check whether last place of the number is 0,2,4,6,8 means number is even. 2. If number is even then write last three digits (100's place, 10's place and 1's place) as a number from the given number. 3. If this number is divisible by 8, then the given number is divisible by 8. If this number is not divisible by 8, then the given number is not divisible by 8. |
For Example:
Let us check whether 9753120 is divisible by 8.
Checking last digit of the number is 0. It means number is even.
Write last three digit number from the given number i.e. 120.
If we divide 120 by 8, then we get quotient 15 along with remainder 0.
So 120 is divisible by 8
So the given number 9753120 is divisible by 8.
Verification: Now divide the number by 8 using conventional division method, we get that the given number 9753120 is divisible by 8 and quotient is 1219140 with remainder 0.
9. Divisible By 9: If sum of all digits in an integer is divisible by 9, then the number is divisible by 9.
| Step for Test: 1. Sum all the digits 2. If sum found is more than one digit number, then again sum all the digits of the previous sum. 3. Repeat the process until we get sum of digits as a one digit number or you can stop up to two digit number less than 90. 4. If final sum is divisible by 9, then the given number is divisible by 9. If final sum is not divisible by 9, then the given number is not divisible by 9. |
For Example:
Let us check whether 162738954 is divisible by 9.
Sum of All digits = 1 + 6 + 2 + 7 + 3 + 8 + 9 + 5 + 4
= 45 (Divisible by 9)
Again Sum all digits = 4 + 5 = 9 (Divisible by 9)
We can see that the sum of all digits in the given number is divisible by 9, so the given number 162738954 is divisible by 9.
Verification: Now divide the number by 3 using conventional division method, we get that the given number 162738954 is divisible by 9 and quotient is 18082106 along with remainder 0.
10. Divisible By 10: If Unit place (Last place) of number has 0 then number is divisible by 10.
| Steps of Test: There is only one step of test i.e. check the last digit is 0. |
For Example: 9450, 348420, 9955770, etc
1. What should be the value of * in 787*838, if number is divisible by 9?
(a) 3 (b) 4 (c) 5 (d) 6
Solution:
Let * place has value a.
If number is divisible by 9 then sum of all digits should be divisible by 9.
7 + 8 + 7 + a + 8 + 3 + 8 = Divisible by 9
41 + a = Divisible by 9
Nearest greater value of multiple of 9 after 41 is 9 × 5 = 45
So 41 + a =45 or a = 45 - 41 or a = 4 (Answer b)
2. The smallest perfect square number, which is divisible by each of 15, 20 and 36 is:
(a) 400 (b) 900 (c) 1600 (d) 2500
Solution:
take LCM of 15, 20 and 36
15 = 3 x 5
20 = 2 x 2 x 5
36 = 2 x 2 x 3 x 3
Now LCM = 2 x 2 x 3 x 3 x 5 = 4 x 9 x 5 = 180
Now read the table of 180 and find the first perfect square
180 x 1 = 180
180 x 2 = 360
180 x 3 = 540
180 x 4 = 720
180 x 5 = 900
So we get 900 as the smallest perfect square. (Answer b)
3. How many numbers between 20 and 451 are divisible by 9?
(a) 45 (b) 46 (c) 47 (d) 48
Solution:
First multiple of 9 after 20 will be 27.
Last multiple of 9 before 451 will be 450.
So the series of numbers divisible by 9 will be (Table of 9 starting with 27)
27, 36, 45, 54, ................................, 450.
So we can see this is an Arithmetic progression term with d = 9, a = 27 and l = 450
We know that nth term of an AP is
l = a + (n - 1) x d
450 = 27 + (n - 1) x 9
or n - 1 = (450 - 27) / 9 =423 / 9 = 47
or n = 47 + 1 = 48
So there are 48 numbers between 20 and 451 which are divisible by 9. (Answer d)
4. Which of Following is not divisible from 4 ?
(a) 7794312 (b) 85547900 (c) 6462602 (d) 62254492
Solution:
For a number to be divisible by 4, last two digits of that number should be divisible by 4
In option (a) Last two digits are 12 which is divisible by 4. So number is also divisible by 4.
In option (b) Last two digits are 00 which is divisible by 4. So number is also divisible by 4.
In option (d) Last two digits are 92 which is divisible by 4. So number is also divisible by 4.
In option (c) Last two digits are 02 which is not divisible by 4.
So number is not divisible by 4. (Answer c)
5. The sum of all 4 digit numbers divisible by 3 is:
(a) 16501500 (b) 18531599 (c) 18533500 (d) 16531399
Solution:
Least four digit number divisible by 3 =1002
Largest four digit number divisible by 3 = 9999
So we have an AP of numbers divisible by 3 as 1002, 1005, 1008, ..............., 9999
To find number of terms n
l = a + (n - 1) x d
9999 = 1002 + (n - 1) x 3
So n - 1 = (9999 - 1002) / 3 = 8997 / 3 = 2999 or n = 2999 + 1 = 3000
Sum of an AP S = (n / 2)(a + l) = 3000 x (9999 + 1002) / 2
or S = 1500 x 11001 = 16501500 (Answer a)
6. Which least digits should come in place of * and @ if the number 4675*2@ is divisible by both 5 & 8?
(a) 1, 0 (b) 4, 5 (c) 7 , 0 (d) 8 , 4
Solution:
For divisibility by 5, last digit should be either 0 or 5.
For divisibility by 8, number should be even at least. so last digit cannot be 5. So, @ = 0
For divisibility by 8, combination of last three digits should be divisible by 8.
If we take * = 0 then last three digit number = 020 (not divisible by 8)
If we take * = 1 then last three digit number = 120 which is divisible by 8.
So, * = 1 and @ = 0 (1, 0) (Answer a)
7. What least number must be added to 3000 to obtain a number exactly divisible by 7?
(a) 2 (b) 3 (c) 4 (d) 5
Solution:
Divide this number and find remainder
7 ) 3000 ( 428
28
20
14
60
56
4
Subtract 7 - 4 = 3
So 3 is the least number which is required to be added to the number to make it divisible by 7. (Answer b)
8. The largest 4 digit number exactly divisible by 4, 9 and 12 is:
(a) 972 (b) 9972 (c) 9988 (d) 988
Solution:
LCM of 4, 9, 12 = 36
Divide largest 4 digit number by 36
36 ) 9999 ( 27
72
279
252
279
252
27
So number exactly divisible by 4, 9, 12 is = 9999 - 27 = 9972 (Answer b)
9. What least number must be subtracted from 5000 to get a number exactly divisible by 23?
(a) 10 (b) 13 (c) 6 (d) 9
Solution:
First divide 5000 by 23 and find remainder. Then subtract the remainder from the given number.
23 ) 5000 ( 217
46
40
23
170
161
9
So 9 should be subtracted from the given number to make it divisible by 23.
10. Find the next number of the series 4, 6, 10, 14, 22, 26, 34, 38, ?
(a) 40 (b) 42 (c) 44 (d) 46
Solution:
See the prime numbers 2 3 5 7 11 13 17 19 23
Double it 4 6 10 14 22 26 34 38 46
so the given series is formed by doubling the prime number.
so next number in the series is 46. (Answer d)
If you feel any problem regarding this topic you can go in comment box and type your problem. i will try to give best solution regarding your problems.
No comments:
Post a Comment