\displaystyle \textbf{Question 1: }\text{Using the test of divisibility, find which of the following numbers are divisible by }3\text{:}
\displaystyle \text{(i) }9572\qquad\text{(ii) }81756\qquad\text{(iii) }258671
\displaystyle \text{(iv) }672588\qquad\text{(v) }105756\qquad\text{(vi) }269784
\displaystyle \text{Answer:}
\displaystyle \text{Note: A number is divisible by }3\text{ if the sum of its digits is divisible by }3.

\displaystyle \text{(i) }9572
\displaystyle 9+5+7+2=23.
\displaystyle \text{Since }23\text{ is not divisible by }3,\ 9572\text{ is not divisible by }3.

\displaystyle \text{(ii) }81756
\displaystyle 8+1+7+5+6=27.
\displaystyle \text{Since }27\text{ is divisible by }3,\ 81756\text{ is divisible by }3.

\displaystyle \text{(iii) }258671
\displaystyle 2+5+8+6+7+1=29.
\displaystyle \text{Since }29\text{ is not divisible by }3,\ 258671\text{ is not divisible by }3.

\displaystyle \text{(iv) }672588
\displaystyle 6+7+2+5+8+8=36.
\displaystyle \text{Since }36\text{ is divisible by }3,\ 672588\text{ is divisible by }3.

\displaystyle \text{(v) }105756
\displaystyle 1+0+5+7+5+6=24.
\displaystyle \text{Since }24\text{ is divisible by }3,\ 105756\text{ is divisible by }3.

\displaystyle \text{(vi) }269784
\displaystyle 2+6+9+7+8+4=36.
\displaystyle \text{Since }36\text{ is divisible by }3,\ 269784\text{ is divisible by }3.
\displaystyle \therefore 81756,\ 672588,\ 105756\text{ and }269784\text{ are divisible by }3.

\displaystyle \textbf{Which of the above numbers are divisible by }9\textbf{?}
\displaystyle \text{Answer:}
\displaystyle \text{Note: A number is divisible by }9\text{ if the sum of its digits is divisible by }9.
\displaystyle \text{For }81756,\quad 8+1+7+5+6=27,\text{ which is divisible by }9.
\displaystyle \text{For }672588,\quad 6+7+2+5+8+8=36,\text{ which is divisible by }9.
\displaystyle \text{For }269784,\quad 2+6+9+7+8+4=36,\text{ which is divisible by }9.
\displaystyle \therefore 81756,\ 672588\text{ and }269784\text{ are divisible by }9.
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Using the test of divisibility, find which of the following numbers are divisible by }3\text{:}
\displaystyle \text{(i) }971234\qquad\text{(ii) }16524\qquad\text{(iii) }382545
\displaystyle \text{(iv) }52618\qquad\text{(v) }843072\qquad\text{(vi) }64128
\displaystyle \text{Which of the above numbers are divisible by }6\text{?}
\displaystyle \text{Answer:}
\displaystyle \text{Note: A number is divisible by }3\text{ if the sum of its digits is divisible by }3.
\displaystyle \text{A number is divisible by }2\text{ if its last digit is }0,\ 2,\ 4,\ 6\text{ or }8.
\displaystyle \text{A number is divisible by }6\text{ if it is divisible by both }2\text{ and }3.

\displaystyle \text{(i) }971234
\displaystyle 9+7+1+2+3+4=26.
\displaystyle \text{Since }26\text{ is not divisible by }3,\ 971234\text{ is not divisible by }3.
\displaystyle \text{Its last digit is }4,\text{ so it is divisible by }2.
\displaystyle \therefore 971234\text{ is not divisible by }6.

\displaystyle \text{(ii) }16524
\displaystyle 1+6+5+2+4=18.
\displaystyle \text{Since }18\text{ is divisible by }3,\ 16524\text{ is divisible by }3.
\displaystyle \text{Its last digit is }4,\text{ so it is also divisible by }2.
\displaystyle \therefore 16524\text{ is divisible by }6.

\displaystyle \text{(iii) }382545
\displaystyle 3+8+2+5+4+5=27.
\displaystyle \text{Since }27\text{ is divisible by }3,\ 382545\text{ is divisible by }3.
\displaystyle \text{Its last digit is }5,\text{ so it is not divisible by }2.
\displaystyle \therefore 382545\text{ is not divisible by }6.

\displaystyle \text{(iv) }52618
\displaystyle 5+2+6+1+8=22.
\displaystyle \text{Since }22\text{ is not divisible by }3,\ 52618\text{ is not divisible by }3.
\displaystyle \text{Its last digit is }8,\text{ so it is divisible by }2.
\displaystyle \therefore 52618\text{ is not divisible by }6.

\displaystyle \text{(v) }843072
\displaystyle 8+4+3+0+7+2=24.
\displaystyle \text{Since }24\text{ is divisible by }3,\ 843072\text{ is divisible by }3.
\displaystyle \text{Its last digit is }2,\text{ so it is also divisible by }2.
\displaystyle \therefore 843072\text{ is divisible by }6.

\displaystyle \text{(vi) }64128
\displaystyle 6+4+1+2+8=21.
\displaystyle \text{Since }21\text{ is divisible by }3,\ 64128\text{ is divisible by }3.
\displaystyle \text{Its last digit is }8,\text{ so it is also divisible by }2.
\displaystyle \therefore 64128\text{ is divisible by }6.
\displaystyle \therefore 16524,\ 382545,\ 843072\text{ and }64128\text{ are divisible by }3.
\displaystyle \therefore 16524,\ 843072\text{ and }64128\text{ are divisible by }6.
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Using the test of divisibility, find which of the following numbers are divisible by }4\text{?}
\displaystyle \text{(i) }5714\qquad\text{(ii) }29546\qquad\text{(iii) }39784
\displaystyle \text{(iv) }64218\qquad\text{(v) }53876\qquad\text{(vi) }736912
\displaystyle \text{Which of the above numbers are divisible by }8\text{?}
\displaystyle \text{Answer:}
\displaystyle \text{Note: A number is divisible by }4\text{ if the number formed by its last two digits is divisible by }4.
\displaystyle \text{A number is divisible by }8\text{ if the number formed by its last three digits is divisible by }8.

\displaystyle \text{(i) }5714
\displaystyle 14\text{ is not divisible by }4.
\displaystyle \therefore 5714\text{ is not divisible by }4.
\displaystyle 714\text{ is not divisible by }8.
\displaystyle \therefore 5714\text{ is not divisible by }8.

\displaystyle \text{(ii) }29546
\displaystyle 46\text{ is not divisible by }4.
\displaystyle \therefore 29546\text{ is not divisible by }4.
\displaystyle 546\text{ is not divisible by }8.
\displaystyle \therefore 29546\text{ is not divisible by }8.

\displaystyle \text{(iii) }39784
\displaystyle 84\text{ is divisible by }4.
\displaystyle \therefore 39784\text{ is divisible by }4.
\displaystyle 784=8\times98.
\displaystyle \therefore 39784\text{ is divisible by }8.

\displaystyle \text{(iv) }64218
\displaystyle 18\text{ is not divisible by }4.
\displaystyle \therefore 64218\text{ is not divisible by }4.
\displaystyle 218\text{ is not divisible by }8.
\displaystyle \therefore 64218\text{ is not divisible by }8.

\displaystyle \text{(v) }53876
\displaystyle 76\text{ is divisible by }4.
\displaystyle \therefore 53876\text{ is divisible by }4.
\displaystyle 876\text{ is not divisible by }8.
\displaystyle \therefore 53876\text{ is not divisible by }8.

\displaystyle \text{(vi) }736912
\displaystyle 12\text{ is divisible by }4.
\displaystyle \therefore 736912\text{ is divisible by }4.
\displaystyle 912=8\times114.
\displaystyle \therefore 736912\text{ is divisible by }8.
\displaystyle \therefore 39784,\ 53876\text{ and }736912\text{ are divisible by }4.
\displaystyle \therefore 39784\text{ and }736912\text{ are divisible by }8.
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Which of the following numbers are divisible by }6\text{?}
\displaystyle \text{(i) }95823\qquad\text{(ii) }723618\qquad\text{(iii) }36912
\displaystyle \text{(iv) }464646\qquad\text{(v) }183627\qquad\text{(vi) }341296
\displaystyle \text{Answer:}
\displaystyle \text{Note: A number is divisible by }3\text{ if the sum of its digits is divisible by }3.
\displaystyle \text{A number is divisible by }2\text{ if its last digit is }0,\ 2,\ 4,\ 6\text{ or }8.
\displaystyle \text{A number is divisible by }6\text{ if it is divisible by both }2\text{ and }3.

\displaystyle \text{(i) }95823
\displaystyle 9+5+8+2+3=27.
\displaystyle \text{Since }27\text{ is divisible by }3,\ 95823\text{ is divisible by }3.
\displaystyle \text{Its last digit is }3,\text{ so it is not divisible by }2.
\displaystyle \therefore 95823\text{ is not divisible by }6.

\displaystyle \text{(ii) }723618
\displaystyle 7+2+3+6+1+8=27.
\displaystyle \text{Since }27\text{ is divisible by }3,\ 723618\text{ is divisible by }3.
\displaystyle \text{Its last digit is }8,\text{ so it is divisible by }2.
\displaystyle \therefore 723618\text{ is divisible by }6.

\displaystyle \text{(iii) }36912
\displaystyle 3+6+9+1+2=21.
\displaystyle \text{Since }21\text{ is divisible by }3,\ 36912\text{ is divisible by }3.
\displaystyle \text{Its last digit is }2,\text{ so it is divisible by }2.
\displaystyle \therefore 36912\text{ is divisible by }6.

\displaystyle \text{(iv) }464646
\displaystyle 4+6+4+6+4+6=30.
\displaystyle \text{Since }30\text{ is divisible by }3,\ 464646\text{ is divisible by }3.
\displaystyle \text{Its last digit is }6,\text{ so it is divisible by }2.
\displaystyle \therefore 464646\text{ is divisible by }6.

\displaystyle \text{(v) }183627
\displaystyle 1+8+3+6+2+7=27.
\displaystyle \text{Since }27\text{ is divisible by }3,\ 183627\text{ is divisible by }3.
\displaystyle \text{Its last digit is }7,\text{ so it is not divisible by }2.
\displaystyle \therefore 183627\text{ is not divisible by }6.

\displaystyle \text{(vi) }341296
\displaystyle 3+4+1+2+9+6=25.
\displaystyle \text{Since }25\text{ is not divisible by }3,\ 341296\text{ is not divisible by }3.
\displaystyle \text{Its last digit is }6,\text{ so it is divisible by }2.
\displaystyle \therefore 341296\text{ is not divisible by }6.
\displaystyle \therefore 723618,\ 36912\text{ and }464646\text{ are divisible by }6.
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Which of the following numbers are divisible by }11\text{?}
\displaystyle \text{(i) }95827\qquad\text{(ii) }110111\qquad\text{(iii) }346929
\displaystyle \text{(iv) }517633\qquad\text{(v) }357269\qquad\text{(vi) }5245185
\displaystyle \text{Answer:}
\displaystyle \text{Note: A number is divisible by }11\text{ if the difference between the sum of its digits}
\displaystyle \text{at odd places and the sum of its digits at even places is }0\text{ or a multiple of }11.

\displaystyle \text{(i) }95827
\displaystyle \text{Difference}=(7+8+9)-(2+5)=24-7=17.
\displaystyle \text{Since }17\text{ is not divisible by }11,\ 95827\text{ is not divisible by }11.

\displaystyle \text{(ii) }110111
\displaystyle \text{Difference}=(1+1+1)-(1+0+1)=3-2=1.
\displaystyle \text{Since }1\text{ is not divisible by }11,\ 110111\text{ is not divisible by }11.

\displaystyle \text{(iii) }346929
\displaystyle \text{Difference}=(9+9+4)-(2+6+3)=22-11=11.
\displaystyle \text{Since }11\text{ is divisible by }11,\ 346929\text{ is divisible by }11.

\displaystyle \text{(iv) }517633
\displaystyle \text{Difference}=(3+6+1)-(3+7+5)=10-15=-5.
\displaystyle \text{Since }-5\text{ is not divisible by }11,\ 517633\text{ is not divisible by }11.

\displaystyle \text{(v) }357269
\displaystyle \text{Difference}=(9+2+5)-(6+7+3)=16-16=0.
\displaystyle \text{Since the difference is }0,\ 357269\text{ is divisible by }11.

\displaystyle \text{(vi) }5245185
\displaystyle \text{Difference}=(5+1+4+5)-(8+5+2)=15-15=0.
\displaystyle \text{Since the difference is }0,\ 5245185\text{ is divisible by }11.
\displaystyle \therefore 346929,\ 357269\text{ and }5245185\text{ are divisible by }11.
\displaystyle \\

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