EXERCISE-2B
Question 1
(i) 2650
(ii) 69435
(iii) 59628
(iv) 789403
(v) 357986
(vi) 367314
Ans.
(i) The given number $=2650$
Digit at unit's place $=0$
$\therefore$ It is divisible by 2
(ii) The given number $=69435$
Digit at unit's place $=5$
$\therefore$ It is not divisible by 2 .
(iii) The given number $=59628$
Digit at unit's place $=8$
$\therefore$ It is divisible by 2 .
(iv) The given number $=789403$
Digit at unit's place $=3$
$\therefore$ It is not divisible by 2 .
(v) The given number $=357986$
Digit at unit's place $=6$
$\therefore$ It is divisible by 2 .
(vi) The given number $=367314$
Digit at unit's place $=4$
$\therefore$ It is divisible by 2 .
Question 2
Q.2. Test the divisibility of following numbers by 3 :
(i) 733
(ii) 10038
(iii) 20701
(iv) 524781
(v) 79124
(vi) 872645
Ans.
(i) The given number $=733$
Sum of its digits $=7+3+3=13$, which is not divisible by 3 .
$\therefore 733$ is not divisible by 3 .
(ii) The given number $=10038$
Sum of its digits $=1+0+0+3+8$
$=12$, which is divisible by 3
$\therefore 10038$ is divisible by 3 .
Sum of its digits $=1+0+0+3+8$
$=12$, which is divisible by 3
$\therefore 10038$ is divisible by 3 .
(iii) The given number $=20701$
Sum of its digits $=2+0+7+0+1$
$=10$, which is not divisible by 3
$\therefore 20701$ is not divisible by 3 .
Sum of its digits $=2+0+7+0+1$
$=10$, which is not divisible by 3
$\therefore 20701$ is not divisible by 3 .
(iv) The given number $=524781$ Sum of its digits $=5+2+4+7+8+1=27$, which is divisible by 3 $\therefore 524781$ is divisible by 3 .
(v) The given number $=79124$ Sum of its digits $=7+9+1+2+4=23$, which is not divisible by 3 $\therefore 79124$ is not divisible by 3 .
(vi) The given number $=872645$ Sum of its digits $=8+7+2+6+4+5=32$, which is not divisible by 3 $\therefore 872645$ is not divisible by 3 .
Question 3
Q.3. Test the divisibility of the follo wing numbers by 4 :
(i) 618
(ii) 2314
(iii) 63712
(iv) 35056
(v) 946126
(vi) 810524
Ans.
(i) The given number $=618$
The number formed by ten's and unit's digits is 18 , which is not divisible by 4 . $\therefore 618$ is not divisible by 4 .
The number formed by ten's and unit's digits is 18 , which is not divisible by 4 . $\therefore 618$ is not divisible by 4 .
(ii) The given number $=2314$
The number formed by ten's and unit's digits is 14 , which is not divisible by 4.
$\therefore 2314$ is not divisible by 4 .
The number formed by ten's and unit's digits is 14 , which is not divisible by 4.
$\therefore 2314$ is not divisible by 4 .
(iii) The given number $=63712$
The number formed by ten's and unit's digits is 12 , which is divisible by 4 $\therefore 63712$ is divisible by 4 .
The number formed by ten's and unit's digits is 12 , which is divisible by 4 $\therefore 63712$ is divisible by 4 .
(iv) The given number $=35056$
The number formed by ten's and unit's digits is 56 , which is divisible by 4 .
$\therefore 35056$ is divisible by 4 .
The number formed by ten's and unit's digits is 56 , which is divisible by 4 .
$\therefore 35056$ is divisible by 4 .
(v) The given number $=946126$
The number formed by ten's and unit's digits is 26 , which is not divisible by 4 . $\therefore 946126$ is not divisible by 4 .
The number formed by ten's and unit's digits is 26 , which is not divisible by 4 . $\therefore 946126$ is not divisible by 4 .
(vi) The given number $=810524$
The number formed by ten's and unit's digits is 24 , which is divisible by 4 . $\therefore 810524$ is divisible by 4 .
Question 4
Question 5
Q.5. Test the divisibility of the follo wing numbers by 6 :
(i) 2070
(ii) 46523
(iii) 71232
(iv) 934706
(v) 251730
(vi) 9087248
Ans.
(i) The given number $=2070$
Its unit's digit $=0$
So, it is divisible by 2
Sum of its digits $=2+0+7+0=9$,
which is divisible by 3
$\therefore$ The given number is divisible by 3 .
So, 2070 is divisible by both 2 and 3.
Hence it is divisible by 6 .
(ii) The given number $=46523$
Its unit's digit = 3
So, it is not divisible by 2
Hence 46523 is not divisible by 6 .
(i) $826,6 \times 2=12$ and 82
Difference between 82 and $12=70$
Which is divisible by 7
$\therefore 826$ is divisible by 7
(ii) $\ln 117,7 \times 2=14,11$
Difference between 14 and 11=14-11=3
Which is not divisible by 7
$\therefore 117$ is not divisible by 7
(iii) $\ln 2345,5 \times 2=10$ and 234
Difference between $234-10=224$
which is divisible by 7
$\therefore 2345$ is divisible by 7
(iv) $\ln 6021,1 \times 2=2$, and 602
Difference between 602 and $2=600$
which is not divisible by 7
$\therefore 6021$ is not divisible by 7
(v) $\operatorname{In} 14126,6 \times 2=12$ and 1412
Difference between $1412-12=1400$
which is divisible by 7
$\therefore 14126$ is divisible by 7
(vi) In $25368,8 \times 2=16$ and 2536
Difference between 2536 and $16=2520$
which is divisible 7
$\therefore 25368$ is divisible by 7
Question 7
Q.7. Test the divisibility of the follo wing numbers by 8 :
(i) 9364
(ii) 2138
(iii) 36792
(iv) 901674
(v) 136976
(vi) 1790184
Ans.
(i) The given number $=9364$
The number formed by hundred's, ten's and unit's digits is 364 , which is not divisible by 8 . $\therefore 9364$ is not divisible by 8 .
(vi) $\ln 326999$
Sum of digits $=3+2+6+9+9+9=38$ which is not divisible by 9 $\therefore 326999$ is divisible by 9
Question 9
Q.9. Test the divisibility of the following number by 10 :
(i) 5790
(ii) 63215
(iii) 55555
Ans. We know that a number is divisible by 10 if its one digit is 0
$\therefore$ (i) 5790 is divisible by 10
Question 10
Q.10. Test the divisibility of the following numbers by 11 :
(i) 4334
(ii) 83721
(iii) 66311
(iv) 137269
(v) 901351
(vi) 8790322
Ans.
(i) The given number $=4334$
Sum of its digits in odd places $=4+3=7$
Sum of its digits in even places $=3+4=7$
Difference of the two sums $=7-7=0$
$\therefore 4334$ is divisible by 11 .
(ii) The given number $=83721$
Sum of its digits in odd places $=1+7+8=16$
Sum of its digits in even places $=2+3=5$
Difference of the two sums $=16-5=11$, which is multiple of 11 .
$\therefore 83721$ is divisible by 11 .
(iii) The given number $=66311$
Sum of its digits in odd places
=1+3+6=10
Sum of its digits in even places
=1+6=7
Difference of the two sums $=10-7=3$, which is not a multiple of 11 .
$\therefore 66311$ is not divisible by 11 .
(iv) The given number $=137269$
Sum of its digits in odd places
=9+2+3=14
Sum of its digits in even places
=6+7+1=14
Difference of the two sums
=14-14=0
$\therefore 137269$ is divisible by 11 .
The number formed by ten's and unit's digits is 24 , which is divisible by 4 . $\therefore 810524$ is divisible by 4 .
Question 4
Question 5
Q.5. Test the divisibility of the follo wing numbers by 6 :
(i) 2070
(ii) 46523
(iii) 71232
(iv) 934706
(v) 251730
(vi) 9087248
Ans.
(i) The given number $=2070$
Its unit's digit $=0$
So, it is divisible by 2
Sum of its digits $=2+0+7+0=9$,
which is divisible by 3
$\therefore$ The given number is divisible by 3 .
So, 2070 is divisible by both 2 and 3.
Hence it is divisible by 6 .
(ii) The given number $=46523$
Its unit's digit = 3
So, it is not divisible by 2
Hence 46523 is not divisible by 6 .
(iii) The given number $=71232$
Its unit's digit = 2
So, it is divisible by 2
Sum of its digits $=7+1+2+3+2$
$=15$, which is divisible by 3
$\therefore 71232$ is divisible by both 2 and 3
Hence it is divisible by 6 .
Its unit's digit = 2
So, it is divisible by 2
Sum of its digits $=7+1+2+3+2$
$=15$, which is divisible by 3
$\therefore 71232$ is divisible by both 2 and 3
Hence it is divisible by 6 .
(iv) The given number $=934706$
Its unit's digit $=6$ So,
it is divisible by 2
Sum of its digits $=9+3+4+7+0+6$
$=29$, which is not divisible by 3
Hence 934706 is not divisible by 6 .
(v) The given number $=251730$
Its unit's digit $=0$
So, it is divisible by 2
Sum of its digits $=2+5+1+7+3+0$
$=18$, which is divisible by 3
$\therefore 251730$ is divisible by both 2 and 3 .
Hence it is divisible by 6 .
(vi) 872536 is not divisible by 6 as sum of its digits is $8+7+2+5+3+6=31$
which is not divisible by 3
Question 6
Q.6. Test the divisibility of the following numbers by 7 :
(i) 826
(ii) 117
(iii) 2345
(iv) 6021
(v) 14126
(vi) 25368
Ans.
We know that a number is divisible by the difference between twice the ones digit and the number formed by the other digits is either 0 or a multiple of 7Its unit's digit $=6$ So,
it is divisible by 2
Sum of its digits $=9+3+4+7+0+6$
$=29$, which is not divisible by 3
Hence 934706 is not divisible by 6 .
(v) The given number $=251730$
Its unit's digit $=0$
So, it is divisible by 2
Sum of its digits $=2+5+1+7+3+0$
$=18$, which is divisible by 3
$\therefore 251730$ is divisible by both 2 and 3 .
Hence it is divisible by 6 .
(vi) 872536 is not divisible by 6 as sum of its digits is $8+7+2+5+3+6=31$
which is not divisible by 3
Question 6
Q.6. Test the divisibility of the following numbers by 7 :
(i) 826
(ii) 117
(iii) 2345
(iv) 6021
(v) 14126
(vi) 25368
Ans.
(i) $826,6 \times 2=12$ and 82
Difference between 82 and $12=70$
Which is divisible by 7
$\therefore 826$ is divisible by 7
(ii) $\ln 117,7 \times 2=14,11$
Difference between 14 and 11=14-11=3
Which is not divisible by 7
$\therefore 117$ is not divisible by 7
(iii) $\ln 2345,5 \times 2=10$ and 234
Difference between $234-10=224$
which is divisible by 7
$\therefore 2345$ is divisible by 7
(iv) $\ln 6021,1 \times 2=2$, and 602
Difference between 602 and $2=600$
which is not divisible by 7
$\therefore 6021$ is not divisible by 7
(v) $\operatorname{In} 14126,6 \times 2=12$ and 1412
Difference between $1412-12=1400$
which is divisible by 7
$\therefore 14126$ is divisible by 7
(vi) In $25368,8 \times 2=16$ and 2536
Difference between 2536 and $16=2520$
which is divisible 7
$\therefore 25368$ is divisible by 7
Question 7
Q.7. Test the divisibility of the follo wing numbers by 8 :
(i) 9364
(ii) 2138
(iii) 36792
(iv) 901674
(v) 136976
(vi) 1790184
Ans.
(i) The given number $=9364$
The number formed by hundred's, ten's and unit's digits is 364 , which is not divisible by 8 . $\therefore 9364$ is not divisible by 8 .
(ii) The given number $=2138$
The number formed by hundred's, ten's and unit's digits is 138 , which is not divisible by 8 .
$\therefore 2138$ is not divisible by 8 .
Question 8
Q.8. Test the divisibility of the following numbers by 9 :
(i) 2358
(ii) 3333
(iii) 98712
(iv) 257106
(v) 647514
(vi) 326999
Ans.
We know that a number is divisible by 9 , if the sum of its digits is divisible by 7
(i) $\ln 2358$
Sum or digits : $2+3+5+8=18$ which is divisible by 9
$\therefore 2358$ is divisible by 9
(ii) $\ln 3333$
Sum of digit $3+3+3+3=12$ which is not divisible by 9
$\therefore 3333$
(iii) $\ln 98712$
Sum of digits $=9+8+7+1+2=27$
Which is divisible by 9
$\therefore 98712$ is divisible by 9
(iv) $\ln 257106$
Sum of digits $=2+5+7+1+0+6=21$ which is not divisible by 9
$\therefore 257106$ is not divisible by 9
(v) $\ln 647514$
Sum of digits $=6+4+7+5+1+4=27$ which is divisible by 9
$\therefore 647514$ is divisible by 9
The number formed by hundred's, ten's and unit's digits is 138 , which is not divisible by 8 .
$\therefore 2138$ is not divisible by 8 .
(iii) The given number $=36792$
The number formed by hundred's, ten's and unit's digits is 792 , which is divisible by 8 .
$\therefore 36792$ is divisible by 8 .
(iv) The given number $=901674$
The number formed by hundred's, ten's and unit's digits is 674 , which is not divisible by 8 .
$\therefore 901674$ is not divisible by 8 .
(v) The given number $=136976$
The number formed by hundred's, ten's and unit's digits is 976 , which is divisible by 8 .
$\therefore 136976$ is divisible by 8 .
(vi) The given number $=1790184$
The number formed by hundred's, ten's and unit's digits is 184 , which is divisible by 8 .
$\therefore 1790184$ is divisible by 8 .
Question 8
(i) 2358
(ii) 3333
(iii) 98712
(iv) 257106
(v) 647514
(vi) 326999
Ans.
We know that a number is divisible by 9 , if the sum of its digits is divisible by 7
(i) $\ln 2358$
Sum or digits : $2+3+5+8=18$ which is divisible by 9
$\therefore 2358$ is divisible by 9
(ii) $\ln 3333$
Sum of digit $3+3+3+3=12$ which is not divisible by 9
$\therefore 3333$
(iii) $\ln 98712$
Sum of digits $=9+8+7+1+2=27$
Which is divisible by 9
$\therefore 98712$ is divisible by 9
(iv) $\ln 257106$
Sum of digits $=2+5+7+1+0+6=21$ which is not divisible by 9
$\therefore 257106$ is not divisible by 9
(v) $\ln 647514$
Sum of digits $=6+4+7+5+1+4=27$ which is divisible by 9
$\therefore 647514$ is divisible by 9
(vi) $\ln 326999$
Sum of digits $=3+2+6+9+9+9=38$ which is not divisible by 9 $\therefore 326999$ is divisible by 9
Question 9
(i) 5790
(ii) 63215
(iii) 55555
Ans. We know that a number is divisible by 10 if its one digit is 0
$\therefore$ (i) 5790 is divisible by 10
Question 10
(i) 4334
(ii) 83721
(iii) 66311
(iv) 137269
(v) 901351
(vi) 8790322
Ans.
(i) The given number $=4334$
Sum of its digits in odd places $=4+3=7$
Sum of its digits in even places $=3+4=7$
Difference of the two sums $=7-7=0$
$\therefore 4334$ is divisible by 11 .
Sum of its digits in odd places $=1+7+8=16$
Sum of its digits in even places $=2+3=5$
Difference of the two sums $=16-5=11$, which is multiple of 11 .
$\therefore 83721$ is divisible by 11 .
(iii) The given number $=66311$
Sum of its digits in odd places
=1+3+6=10
Sum of its digits in even places
=1+6=7
Difference of the two sums $=10-7=3$, which is not a multiple of 11 .
$\therefore 66311$ is not divisible by 11 .
(iv) The given number $=137269$
Sum of its digits in odd places
=9+2+3=14
Sum of its digits in even places
=6+7+1=14
Difference of the two sums
=14-14=0
$\therefore 137269$ is divisible by 11 .
(v) The given number $=901351$
Sum of its digits in odd places
=1+3+0=4
Sum of its digits in even places
=5+1+9=15
Difference of the two sums $=15-4$
$=11$, which is a multiple of 11 .
$\therefore 901351$ is divisible by 11 .
Sum of its digits in odd places
=1+3+0=4
Sum of its digits in even places
=5+1+9=15
Difference of the two sums $=15-4$
$=11$, which is a multiple of 11 .
$\therefore 901351$ is divisible by 11 .
(vi) The given number $=8790322$
Sum of its digits in odd places
=2+3+9+8=22
Sum of its digits in even places
=2+0+7=9
Difference of the two sums
=22-9=13
which is not a multiple of 11 .
$\therefore 8790322$ is not divisible by 11 .
Question 11
Q.11. In each of the following numbers, replace * by the smallest number to make it divisible by 3 .
(i) $27 * 4$
(ii) $53 * 46$
(iii) $8 * 711$
(iv) $62 * 35$
(v) $234 * 17$
(vi) 6*1054
Ans.
(i) The given number $=27 * 4$
Sum of its digits $=2+7+4=13$
The number next to 13 which is divisible by 3 is 15 .
$\therefore$ Required smallest number $=15-13$
=2
(ii) The given number $=53^* 46$
Sum of the given digits $=5+3+4+6$
$=18$, which is divisible by 3 .
$\therefore$ Required smallest number $=0$.
(iii) The given number $=8 * 711$
Sum of the given digits $=8+7+1+1=17$
The number next to 17 , which is divisible by 3 is 18 .
$
\begin{aligned}
& \therefore \text { Required smallest number }=18-17 \\
& =1
\end{aligned}
$
(iv) The given number $=62 * 35$
Sum of the given digits $=6+2+3+5$
=16
The number next to 16 , which is divisible by 3 is 18 .
$
\begin{aligned}
& \therefore \text { Required smallest number }=18-16 \\
& =2
\end{aligned}
$
(v) The given number $=234 * 17$
Sum of the given digits
=2+3+4+1+7=17
The number next to 17 , which is divisible by 3 is 18 .
$\therefore$ Required smallest number
$
=18-17=1 \text {. }
$
(vi) The given number $=6 * 1054$
Sum of the given digits $=6+1+0+5+4=16$
The number next to 16 , which is divisible by 3 is 18 .
$\therefore$ Required smallest number
=18-16=2 .
Question 12
Q.12. In each of the following numbers, replace * by the smallest number to make it divisible by 9 .
(i) $65 * 5$
(ii) 2*135
(iii) 6702*
(iv) $91 * 67$
(v) 6678*1
(vi) $835 * 86$
Ans.
(i) The given number $=65 * 5$
Sum of its given digits $=6+5+5=16$
The number next to 16 , which is divisible by 9 is 18 .
$
\begin{aligned}
& \therefore \text { Required smallest number }=18-16 \\
& =2
\end{aligned}
$
(ii) The given number $=2 * 135$
Sum of its given digits $=2+1+3+5$
=11
The number next to 11 , which is divisible by 9 is 18 .
$\therefore$ Required smallest number
$
=18-11=7 \text {. }
$
(iii) The given number $=6702^*$
Sum of its given digits
=6+7+0+2=15
The number next to 15 , which is divisible by 9 is 18 .
$\therefore$ Required smallest number $=18-15=3$
Sum of its digits $=2+7+4=13$
The number next to 13 which is divisible by 3 is 15 .
$\therefore$ Required smallest number $=15-13$
=2
(ii) The given number $=53^* 46$
Sum of the given digits $=5+3+4+6$
$=18$, which is divisible by 3 .
$\therefore$ Required smallest number $=0$.
(iii) The given number $=8 * 711$
Sum of the given digits $=8+7+1+1=17$
The number next to 17 , which is divisible by 3 is 18 .
$
\begin{aligned}
& \therefore \text { Required smallest number }=18-17 \\
& =1
\end{aligned}
$
(iv) The given number $=62 * 35$
Sum of the given digits $=6+2+3+5$
=16
The number next to 16 , which is divisible by 3 is 18 .
$
\begin{aligned}
& \therefore \text { Required smallest number }=18-16 \\
& =2
\end{aligned}
$
(v) The given number $=234 * 17$
Sum of the given digits
=2+3+4+1+7=17
The number next to 17 , which is divisible by 3 is 18 .
$\therefore$ Required smallest number
$
=18-17=1 \text {. }
$
(vi) The given number $=6 * 1054$
Sum of the given digits $=6+1+0+5+4=16$
The number next to 16 , which is divisible by 3 is 18 .
$\therefore$ Required smallest number
=18-16=2 .
Question 12
(i) $65 * 5$
(ii) 2*135
(iii) 6702*
(iv) $91 * 67$
(v) 6678*1
(vi) $835 * 86$
Ans.
(i) The given number $=65 * 5$
Sum of its given digits $=6+5+5=16$
The number next to 16 , which is divisible by 9 is 18 .
$
\begin{aligned}
& \therefore \text { Required smallest number }=18-16 \\
& =2
\end{aligned}
$
(ii) The given number $=2 * 135$
Sum of its given digits $=2+1+3+5$
=11
The number next to 11 , which is divisible by 9 is 18 .
$\therefore$ Required smallest number
$
=18-11=7 \text {. }
$
(iii) The given number $=6702^*$
Sum of its given digits
=6+7+0+2=15
The number next to 15 , which is divisible by 9 is 18 .
$\therefore$ Required smallest number $=18-15=3$
(iv) The given number $=91 * 67$
Sum of its given digits $=9+1+6+7=23$
The number next to 23 , which is divisible by 9 is 27 .
$\therefore$ Required smallest number $=27-23=4$.
Sum of its given digits $=9+1+6+7=23$
The number next to 23 , which is divisible by 9 is 27 .
$\therefore$ Required smallest number $=27-23=4$.
(v) The given number $=6678 * 1$
Sum of its given digits
=6+6+7+8+1=28
The number next to 28 , which is divisible by 9 is 36 .
$\therefore$ Required smallest number
=36-28=8 .
(vi) The given number $=835 * 86$
Sum of its given digits
$
\begin{aligned}
& =8+3+5+8+6 \\
& =30
\end{aligned}
$
The number next to 30 , which is divisible by 9 is 36 .
$\therefore$ Required smallest number
=36-30=6 .
Question 13
Q.13. In each of the following numbers, replace * by the smallest number to make it divisible by 11.
(i) $26 * 5$
(ii) $39 * 43$
(iii) $86 * 72$
(iv) $467^* 91$
(v) $1723 * 4$
(vi) 9*8071
Ans.
(iii) The given number $=86^* 72$
Sum of its digits in odd places
=2+*+8=*+10
Sum of its digits in even places
=7+6=13
Difference of the two sums
=*+10-13=^*-3
The given number will be divisible by 11, if the difference of the two sums $=0$.
$
\begin{aligned}
& \therefore^*-3=0 \\
& *=3
\end{aligned}
$
$\therefore$ Required smallest number =3
Sum of its given digits
=6+6+7+8+1=28
The number next to 28 , which is divisible by 9 is 36 .
$\therefore$ Required smallest number
=36-28=8 .
(vi) The given number $=835 * 86$
Sum of its given digits
$
\begin{aligned}
& =8+3+5+8+6 \\
& =30
\end{aligned}
$
The number next to 30 , which is divisible by 9 is 36 .
$\therefore$ Required smallest number
=36-30=6 .
Question 13
Q.13. In each of the following numbers, replace * by the smallest number to make it divisible by 11.
(i) $26 * 5$
(ii) $39 * 43$
(iii) $86 * 72$
(iv) $467^* 91$
(v) $1723 * 4$
(vi) 9*8071
Ans.
(i) The given number $=26 * 5$
Sum of its digits is odd places
=5+6=11
Sum of its digits in even places $=*+2$
Difference of the two sums
=11-(*+2)
The given number will be divisible by 11 if the difference of the two sums $=0$.
$
\begin{aligned}
& \therefore 11-\left(^*+2\right)=0 \\
& 11=*+2 \\
& 11-2=* \\
& 9=*
\end{aligned}
$
$\therefore$ Required smallest number $=9$.
Sum of its digits is odd places
=5+6=11
Sum of its digits in even places $=*+2$
Difference of the two sums
=11-(*+2)
The given number will be divisible by 11 if the difference of the two sums $=0$.
$
\begin{aligned}
& \therefore 11-\left(^*+2\right)=0 \\
& 11=*+2 \\
& 11-2=* \\
& 9=*
\end{aligned}
$
$\therefore$ Required smallest number $=9$.
(ii) The given number $=39 * 43$
Sum of its digits in odd places
=3+*+3=*+6
Sum of its digits in even places
=4+9=13
Difference of the two sums
=*+6-13=*-7
The given number will be divisible by 11 , if the difference of the two sums $=0$.
$
\begin{aligned}
& \therefore^*-7=0 \\
& *=7
\end{aligned}
$
$\therefore$ Required smallest number =7
Sum of its digits in odd places
=3+*+3=*+6
Sum of its digits in even places
=4+9=13
Difference of the two sums
=*+6-13=*-7
The given number will be divisible by 11 , if the difference of the two sums $=0$.
$
\begin{aligned}
& \therefore^*-7=0 \\
& *=7
\end{aligned}
$
$\therefore$ Required smallest number =7
(iii) The given number $=86^* 72$
Sum of its digits in odd places
=2+*+8=*+10
Sum of its digits in even places
=7+6=13
Difference of the two sums
=*+10-13=^*-3
The given number will be divisible by 11, if the difference of the two sums $=0$.
$
\begin{aligned}
& \therefore^*-3=0 \\
& *=3
\end{aligned}
$
$\therefore$ Required smallest number =3
(iv) The given number $=467^* 91$
Sum of its digits in odd places
=1+*+6=*+7
Sum of its digits in even places
=9+7+4=20
Difference of the two sums
$
\begin{aligned}
& =20-(*+7) \\
& =20-*-7=13-*
\end{aligned}
$
Clearly the difference of the two sums will be multiple of 11 if $13-*=11$
$
\begin{aligned}
& \therefore 13-11=* \\
& 2=* \\
& *=2
\end{aligned}
$
$\therefore$ Required smallest number =2
(v) The given number $=1723 * 4$
Sum of its digits in odd places
=4+3+7=14
Sum of its digits in even places
=*+2+1=*+3
Difference of the two sums
=*+3-14=*-11
$\therefore$ The given number will be divisible by 11 , if * -11 is a multiple of 11 , which is possible if * 0
$\therefore$ Required smallest number=0
(vi) The given number $=9 * 8071$
Sum of its digits in odd places
=1+0+*=1+*
Sum of its digits in even places
=7+8+9=24
Difference of the two sums
=24-1-^*=23-*
$\therefore$ The given number will be divisible by
11 , if $23-*$ is a multiple of 11 , which is possible if $*=1$.
$\therefore$ Required smallest number =1
Question 14
Q.14. Test the divisibility of
(i) 10000001 by 11
(ii) 19083625 by 11
(iii) 2134563 by 9
(iv) 10001001 by 3
(v) 10203574 by 4
(vi) 12030624 by 8
Ans.
(i) The given number =10000001
Sum of its digits in odd places
=1+0+0+0=1
Sum of its digits in even places
=0+0+0+1=1
Difference of the two sums =1-1=0
$\therefore$ The number 10000001 is divisible by 11 .
(ii) The given number =19083625
Sum of its digits in odd places
=5+6+8+9=28
Sum of its digits in even places
=2+3+0+1=6
Difference of the two sums =28-6
=22, which is divisible by 11 .
$\therefore$ The number 19083625 is divisible by 11 .
(vi) The given number =12030624
The number formed by its hundred's, ten's and unit's digits $=624$, which is divisible by 8 .
$\therefore$ The number 12030624 is divisible by 8 .
Question 15
Q.15. Which of the following are prime numbers?
(i) 103
(ii) 137
(iii) 161
(iv) 179
(v) 217
(vi) 277
(vii) 331
(viii) 397
Ans.
103,137,179,277,331,397 are prime numbers.
Question 16
Q.16. Give an example of a number
(i) Which is divisible by 2 but not by 4 .
(ii) Which is divisible by 4 but not by 8 .
(iii) Which is divisible by both 2 and 8 but not divisible by 16 .
(v) Which is divisible by both 3 and 6 but not by 18.
Ans.
(i) 154
(ii) 612
(iii) 5112,3816
(iv) 3426,5142 etc.
Question 17
Q.17. Write (T) for true and (F) for false against each of the followings statements :
(i) If a number is divisible by 4 , it must be divisible by 8 .
(ii) If a number is divisible by 8 , it must be divisible by 4 .
(iii) If a number divides the sum of two numbers exactly, it must exactly divide the numbers separately.
(iv) If a number is divisible by both 9 and 10, it must be divisible by 90 .
(v) A number is divisible by 18 , if it is divisible by both 3 and 6.
(vi) If a number is divisible by 3 and 7 , it must be divisible by 21.
(vii) The sum of two consecutive odd numbers is always divisible by 4.
(viii) If a number divides three numbers exactly, it must divide their sum exactly.
Ans.
(i) False
(ii) True
(iii) False
(iv) True
(v) False
(vi) True
(vii) True
(viii) True.
PRIME FACTORIZATION
Prime Factor. A factor of a given number is called a prime factor if this factor is a prime number.
Prime Factorization. To express a given number as a product of prime factors is called a prime factorization of the given number.
Sum of its digits in odd places
=1+0+*=1+*
Sum of its digits in even places
=7+8+9=24
Difference of the two sums
=24-1-^*=23-*
$\therefore$ The given number will be divisible by
11 , if $23-*$ is a multiple of 11 , which is possible if $*=1$.
$\therefore$ Required smallest number =1
Question 14
Q.14. Test the divisibility of
(i) 10000001 by 11
(ii) 19083625 by 11
(iii) 2134563 by 9
(iv) 10001001 by 3
(v) 10203574 by 4
(vi) 12030624 by 8
Ans.
(i) The given number =10000001
Sum of its digits in odd places
=1+0+0+0=1
Sum of its digits in even places
=0+0+0+1=1
Difference of the two sums =1-1=0
$\therefore$ The number 10000001 is divisible by 11 .
(ii) The given number =19083625
Sum of its digits in odd places
=5+6+8+9=28
Sum of its digits in even places
=2+3+0+1=6
Difference of the two sums =28-6
=22, which is divisible by 11 .
$\therefore$ The number 19083625 is divisible by 11 .
(iii) The given number =2134563
Sum of its digits =2+1+3+4+5+6+3
=24, which is not divisible by 9 .
$\therefore$ The number 2134563 is not divisible by 9 .
Sum of its digits =2+1+3+4+5+6+3
=24, which is not divisible by 9 .
$\therefore$ The number 2134563 is not divisible by 9 .
(iv) The given number =10001001
Sum of its digits =1+0+0+0+1+0+0+1=3, which is divisible by 3 .
$\therefore$ The number 10001001 is divisible by 3 .
Sum of its digits =1+0+0+0+1+0+0+1=3, which is divisible by 3 .
$\therefore$ The number 10001001 is divisible by 3 .
(v) The given number =10203574
The number formed by its ten's and unit's digits is 74 , which is not divisible by 4 .
$\therefore$ The number 10203574 is not divisible by 4 .
The number formed by its ten's and unit's digits is 74 , which is not divisible by 4 .
$\therefore$ The number 10203574 is not divisible by 4 .
(vi) The given number =12030624
The number formed by its hundred's, ten's and unit's digits $=624$, which is divisible by 8 .
$\therefore$ The number 12030624 is divisible by 8 .
Question 15
Q.15. Which of the following are prime numbers?
(i) 103
(ii) 137
(iii) 161
(iv) 179
(v) 217
(vi) 277
(vii) 331
(viii) 397
Ans.
103,137,179,277,331,397 are prime numbers.
Question 16
Q.16. Give an example of a number
(i) Which is divisible by 2 but not by 4 .
(ii) Which is divisible by 4 but not by 8 .
(iii) Which is divisible by both 2 and 8 but not divisible by 16 .
(v) Which is divisible by both 3 and 6 but not by 18.
Ans.
(i) 154
(ii) 612
(iii) 5112,3816
(iv) 3426,5142 etc.
Question 17
(i) If a number is divisible by 4 , it must be divisible by 8 .
(ii) If a number is divisible by 8 , it must be divisible by 4 .
(iii) If a number divides the sum of two numbers exactly, it must exactly divide the numbers separately.
(iv) If a number is divisible by both 9 and 10, it must be divisible by 90 .
(v) A number is divisible by 18 , if it is divisible by both 3 and 6.
(vi) If a number is divisible by 3 and 7 , it must be divisible by 21.
(vii) The sum of two consecutive odd numbers is always divisible by 4.
(viii) If a number divides three numbers exactly, it must divide their sum exactly.
Ans.
(i) False
(ii) True
(iii) False
(iv) True
(v) False
(vi) True
(vii) True
(viii) True.
PRIME FACTORIZATION
Prime Factor. A factor of a given number is called a prime factor if this factor is a prime number.
Prime Factorization. To express a given number as a product of prime factors is called a prime factorization of the given number.
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