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)  $269784$

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?

• 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 $8$?

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)  $341296$

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)  $5245185$

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 $11$