Q1. Using the test of divisibility, find which of the following numbers are divisible by 3?

1. $9572$
2. $81756$
3. $258671$
4. $672588$
5. $105756$
6. $269784$

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

1. $81756$
• Sum of $8 + 1 + 7 + 5 + 6 = 27. 27$ is divisible by $3$. Hence $81756$ is divisible by $3$.
2. $258671$
• Sum of $2 + 5 + 8 + 6 + 7 + 1 = 29$. $29$ is not divisible by $3$. Hence $258671$ is divisible by $3$.
3. $672588$
• Sum of $6 + 7 + 2 + 5 + 8 + 8 = 36$. $36$ is divisible by $3$. Hence $672588$ is divisible by $3$.
4. $105756$
• Sum of $1 + 0 + 5 + 7 + 5 + 6 = 24$. $24$ is divisible by $3$. Hence $105756$ is divisible by $3$.
5. $269784$
• 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 $9$.

Q2. Using the test of divisibility, find which of the following numbers are divisible by 3?

1. $971234$
2. $16524$
3. $382545$
4. $52618$
5. $843072$
6. $64128$

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 $3$.
1. $971234$
• 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 $2$.
2. $16524$
• Sum of $1 + 6 + 5 + 2 + 4 = 18. 18$ is divisible both by $2$ and $3$. Hence the number is also divisible by $6$.
3. $382545$
• 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 $6$.
4. $52618$
• Sum of $5 + 2 + 6 + 1 + 8 = 22. 22$ is divisible by $2$ but not by $3$. Hence the number is not divisible by $6$.
5. $843072$
• Sum of $8 + 4 + 3 + 0 + 7 + 2 = 24. 24$ is divisible by $2$ and $3$. Hence the number is also divisible by $6$.
6. $64128$
• Sum of $6 + 4 + 1 + 2 + 8 = 21. 21$ is divisible by $3$ but not by $2$. Hence the number is not divisible by $6$.

Q3. Using the test of divisibility, find which of the following numbers are divisible by 4?

1. $5714$
2. $29546$
3. $39784$
4. $64218$
5. $53876$
6. $736912$

Which of the above numbers are divisible by \$latex 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 $8$
1. $5714$
• $14$ 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 $8$.
2. $29546$
• $46$ 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 $8$.
3. $39784$
• $84$ is divisible by $4$. Hence the $39784$ is divisible by $4$.
• $784$ is divisible by $8$. Hence the $39784$ is divisible by $8$.
4. $64218$
• $18$ 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 $8$.
5. $53876$
• $76$ is divisible by $4$. Hence the $53876$ is divisible by $4$.
• $876$ is divisible by $8$. Hence the $53876$ is divisible by $8$.
6. $736912$
• $12$ is divisible by $4$. Hence the $736912$ is divisible by $4$.
• $912$ is divisible by $8$. Hence the $736912$ is divisible by $8$.

Q4. Which of the above numbers are divisible by 6?

1. $95823$
2. $723618$
3. $36912$
4. $464646$
5. $183627$
6. $341296$

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 $3$.
1. $95823$
• 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 $6$.
2. $723618$
• 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 $6$.
3. $36912$
• 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 $6$.
4. $464646$
• 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 $6$.
5. $183627$
• 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 $6$.
6. $341296$
• 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 $6$.

Q5. Which of the above numbers are divisible by $11$?

1. $95827$
2. $110111$
3. $346929$
4. $517633$
5. $357269$
6. $5245185$

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 $11$.

1. $95827$
• 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 $11$.
2. $110111$
• 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 $11$.
3. $346929$
• 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 $11$.
4. $517633$
• 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 $11$.
5. $357269$
• 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 $11$.
6. $5245185$
• 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 $11$