Class 8: Factors & Multiples – Exercise 4A – ICSE / ISC / CBSE Mathematics Portal for K12 Students

Question 1: Using the test of divisibility, find which of the following numbers are divisible by 3?

(i) $9572$ (ii) $81756$ (iii) $258671$ (iv) $672588$ (v) $105756$ (vi) $269784$

Answer:

Note: A number is divisible by $3$ , if the sum of the digits of the number is divisible by $3$ .

(i) $9572$

Sum of $9 + 5 + 7 + 2 = 24. 24$ is divisible by $3$ . Hence $9572$ is divisible by $3$ .

(ii) $81756$

Sum of $8 + 1 + 7 + 5 + 6 = 27. 27$ is divisible by $3$ . Hence $81756$ is divisible by $3$ .

(iii) $258671$

Sum of $2 + 5 + 8 + 6 + 7 + 1 = 29$ . $29$ is not divisible by $3$ . Hence $258671$ is divisible by $3$ .

(iv) $672588$

Sum of $6 + 7 + 2 + 5 + 8 + 8 = 36$ . $36$ is divisible by $3$ . Hence $672588$ is divisible by $3$ .

(v) $105756$

Sum of $1 + 0 + 5 + 7 + 5 + 6 = 24$ . $24$ is divisible by $3$ . Hence $105756$ is divisible by $3$ .

(vi) $269784$

Sum of $2 + 6 + 9 + 7 + 8 + 4 = 36$ . $36$ is divisible by $3$ . Hence $269784$ is divisible by $3$ .

Which of the above numbers are divisible by 9?

Note: A number is divisible by $9$ if the sum of the digits of the number is divisible by $9$ .

Therefore, $81756, 672588$ and $269784$ are divisible by $9$ .

$\\$

Question 2: Using the test of divisibility, find which of the following numbers are divisible by 3?

(i) $971234$ (ii) $16524$ (iii) $382545$ (iv) $52618$ (v) $843072$ (vi) $64128$

Which of the above numbers are divisible by 6?

Answer:

Note: A number is divisible by $3$ , if the sum of the digits of the number is divisible by $3$ .
Note: A number is divisible by $2$ , if the last digit of the number is $0, 2, 4, 6$ , and $8$ .
Note: A number is divisible by $6$ , if it is divisible both by $2$ and $3$ .

(i) $971234$

Sum of $9 + 7 + 1 + 2 + 3 + 4 = 26$ . 26 $ is not divisible by $3$ . Hence $971234$ is not divisible by $3$ . It is divisible by $2$ .

(ii) $16524$

Sum of $1 + 6 + 5 + 2 + 4 = 18. 18$ is divisible both by $2$ and $3$ . Hence the number is also divisible by $6$ .

(iii) $382545$

Sum of $3 + 8 + 2 + 5 + 4 + 5 = 27. 27$ is divisible by $3$ but not by $2$ . Hence the number is not divisible by $6$ .

(iv) $52618$

Sum of $5 + 2 + 6 + 1 + 8 = 22. 22$ is divisible by $2$ but not by $3$ . Hence the number is not divisible by $6$ .

(v) $843072$

Sum of $8 + 4 + 3 + 0 + 7 + 2 = 24. 24$ is divisible by $2$ and $3$ . Hence the number is also divisible by $6$ .

(vi) $64128$

Sum of $6 + 4 + 1 + 2 + 8 = 21. 21$ is divisible by $3$ but not by $2$ . Hence the number is not divisible by $6$ .

$\\$

Question 3: Using the test of divisibility, find which of the following numbers are divisible by $4$ ?

(i) $5714$ (ii) $29546$ (iii) $39784$ (iv) $64218$ (v) $53876$ (vi) $736912$

Which of the above numbers are divisible by $8$ ?

Answer:

Note:

A number is divisible by $4$ if the number formed by the last two digits of the number is divisible by $4$ .
A number is divisible by $8$ if the number formed by then last three digits of the number is divisible by $8$

(i) $5714$

$14$ is not divisible by $4$ . Hence the $5714$ is not divisible by $4$ .

$714$ is not divisible by $8$ . Hence the $5714$ is not divisible by $8$ .

(ii) $29546$

$46$ is not divisible by $4$ . Hence the $29546$ is not divisible by $4$ .

$546$ is not divisible by $8$ . Hence the $29546$ is not divisible by $8$ .

(iii) $39784$

$84$ is divisible by $4$ . Hence the $39784$ is divisible by $4$ .

$784$ is divisible by $8$ . Hence the $39784$ is divisible by $8$ .

(iv) $64218$

$18$ is not divisible by $4$ . Hence the $64218$ is not divisible by $4$ .

$218$ is not divisible by $8$ . Hence the $64218$ is not divisible by $8$ .

(v) $53876$

$76$ is divisible by $4$ . Hence the $53876$ is divisible by $4$ .

$876$ is divisible by $8$ . Hence the $53876$ is divisible by $8$ .

(vi) $736912$

$12$ is divisible by $4$ . Hence the $736912$ is divisible by $4$ .

$912$ is divisible by $8$ . Hence the $736912$ is divisible by $8$ .

$\\$

Question 4: Which of the above numbers are divisible by 6?

(i) $95823$ (ii) $723618$ (iii) $36912$ (iv) $464646$ (v) $183627$ (vi) $341296$

Answer:

Note:

A number is divisible by $3$ , if the sum of the digits of the number is divisible by $3$ .
A number is divisible by $2$ , if the last digit of the number is $0, 2, 4, 6$ , and $8$ .
A number is divisible by $6$ , if it is divisible both by $2$ and $3$ .

(i) $95823$

Sum of $9 + 5 + 8 + 2 + 3 = 27. 27$ is divisible by $3$ . Hence $95823$ is divisible by $3$ .

The last digit is $3$ which is not divisible by $2$ .

Hence the number is not divisible by $6$ .

(ii) $723618$

Sum of $7 + 2 + 3 + 6 + 1 + 8 = 27. 27$ is divisible by $3$ . Hence $723618$ is divisible by $3$ .

The last digit is $8$ which is divisible by $2$ .

Hence the number is divisible by $6$ .

(iii) $36912$

Sum of $3 + 6 + 9 + 1 + 2 = 21. 21$ is divisible by $3$ . Hence $36912$ is divisible by $3$ .

The last digit is $2$ which is divisible by $2$ .

Hence the number is divisible by $6$ .

(iv) $464646$

Sum of $4 + 6 + 4 + 6 + 4 + 6 = 30. 30$ is divisible by $3$ . Hence $464646$ is divisible by $3$ .

The last digit is $6$ which is divisible by $2$ .

Hence the number is divisible by $6$ .

(v) $183627$

Sum of $1 + 8 + 3 + 6 + 2 + 7 = 27. 27$ is divisible by $3$ . Hence $183627$ is divisible by $3$ .

The last digit is $7$ which is not divisible by $2$ .

Hence the number is not divisible by $6$ .

(vi) $341296$

Sum of $3 + 4 + 1 + 2 + 9 + 6 = 25. 27$ is not divisible by $3$ . Hence $341296$ is divisible by $3$ .

The last digit is $3$ which is not divisible by $2$ .

Hence the number is not divisible by $6$ .

Question 5: Which of the above numbers are divisible by $11$ ?

(i) $95827$ (ii) $110111$ (iii) $346929$ (iv) $517633$ (v) $357269$ v(i) $5245185$

Answer:

Note: A number is divisible by $11$ id the difference between the sum of its digits at odd places and the sum of the digits at even places is either 0 or a number divisible by $11$ .

(i) $95827$

Difference = (sum of digits at odd places) – (sum of digits at even places) $= (7 + 8 + 9) - (2 + 5) = 24 - 7 = 17$ .

$17$ is not divisible by $11$ . Hence then number is not divisible by $11$ .

(ii) $110111$

Difference = (sum of digits at odd places) – (sum of digits at even places) $= (1 + 1 + 1) - (1+0 +1) = 3 - 1 = 1$ .

$1$ is not divisible by $11$ . Hence then number is not divisible by $11$ .

(iii) $346929$

Difference = (sum of digits at odd places) – (sum of digits at even places) $= (9 + 9 + 4) - (2 + 6 + 3) = 22 - 11=11$

$11$ is divisible by $11$ . Hence then number is divisible by $11$ .

(iv) $517633$

Difference = (sum of digits at odd places) – (sum of digits at even places) $(3 + 6 + 1) - (3 + 7 + 5) = 10 - 15 = -5$

$-5$ is no divisible by $11$ . Hence then number is not divisible by $11$ .

(v) $357269$

Difference = (sum of digits at odd places) – (sum of digits at even places) $(9 + 2 + 5) - (6 + 7 + 3) = 16 - 16 = 0$

Hence then number is divisible by $11$ .

(vi) $5245185$

Difference = (sum of digits at odd places) – (sum of digits at even places) $(5 + 1 + 4 + 5) - (8 + 5 + 2) = 15 - 15 = 0$

Hence then number is divisible by $11$ .

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Class 8: Factors & Multiples – Exercise 4A

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