EXERCISE-2C
Question 1
Find the prime factorization of each of the following numbers.
12
Ans. We have
\begin{array}{l|r}
2 & 12 \\
\hline 2 & 6 \\
\hline 3 & 3 \\
\hline & 1
\end{array}
$
$\therefore 12=2 \times 2 \times 3=2^2 \times 3 \text {. }$
Question 2
Sol :
We have
$\begin{array}{l|c}2 & 18 \\
\hline 3 & 9 \\
\hline 3 & 3 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 18=2 \times 3 \times 3 \\
& =2 \times 3^2 .
\end{aligned}$
Question 3
48
Ans. We have
$\begin{array}{l|r}
2 & 48 \\
\hline 2 & 24 \\
\hline 2 & 12 \\
\hline 2 & 6 \\
\hline 3 & 3 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 48=2 \times 2 \times 2 \times 2 \times 3 \\
& =2^4 \times 3
\end{aligned}
$
Question 456.
Ans.
$\begin{array}{l|r}
2 & 48 \\
\hline 2 & 24 \\
\hline 2 & 12 \\
\hline 2 & 6 \\
\hline 3 & 3 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 48=2 \times 2 \times 2 \times 2 \times 3 \\
& =2^4 \times 3
\end{aligned}
$
Question 4
Ans.
We have
$\begin{array}{l|r}
2 & 56 \\
\hline 2 & 28 \\
\hline 2 & 14 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 56=2 \times 2 \times 2 \times 7 \\
& =2^3 \times 7 .
\end{aligned}
$
Question 590
Ans.
Question 9
637
Ans.
Question 11
1224
Ans.
$\begin{array}{l|r}
2 & 56 \\
\hline 2 & 28 \\
\hline 2 & 14 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 56=2 \times 2 \times 2 \times 7 \\
& =2^3 \times 7 .
\end{aligned}
$
Question 5
Ans.
We have
$\begin{array}{l|r}
2 & 90 \\
\hline 3 & 45 \\
\hline 3 & 15 \\
\hline 5 & 5 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 90=2 \times 3 \times 3 \times 5 \\
& =2 \times 3^2 \times 5
\end{aligned}
$
Question 6136
Ans.
$\begin{array}{l|r}
2 & 90 \\
\hline 3 & 45 \\
\hline 3 & 15 \\
\hline 5 & 5 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 90=2 \times 3 \times 3 \times 5 \\
& =2 \times 3^2 \times 5
\end{aligned}
$
Question 6
Ans.
We have
$\begin{array}{r|r}
2 & 136 \\
\hline 2 & 68 \\
\hline 2 & 34 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 136=2 \times 2 \times 2 \times 17 \\
& =2^3 \times 17
\end{aligned}
$
Question 7
252
Ans.
$\begin{array}{r|r}
2 & 136 \\
\hline 2 & 68 \\
\hline 2 & 34 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 136=2 \times 2 \times 2 \times 17 \\
& =2^3 \times 17
\end{aligned}
$
Question 7
Ans.
We have
$\begin{array}{l|r}
2 & 252 \\
\hline 2 & 126 \\
\hline 3 & 63 \\
\hline 3 & 21 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 252=2 \times 2 \times 3 \times 3 \times 7 \\
& =2^2 \times 3^2 \times 7
\end{aligned}
$
Question 8
420
Ans.
$\begin{array}{l|r}
2 & 252 \\
\hline 2 & 126 \\
\hline 3 & 63 \\
\hline 3 & 21 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 252=2 \times 2 \times 3 \times 3 \times 7 \\
& =2^2 \times 3^2 \times 7
\end{aligned}
$
Question 8
Ans.
We have
$\begin{array}{r|r}
2 & 420 \\
\hline 2 & 210 \\
\hline 3 & 105 \\
\hline 5 & 35 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 420=2 \times 2 \times 3 \times 5 \times 7 \\
& =2^2 \times 3 \times 5 \times 7
\end{aligned}
$
$\begin{array}{r|r}
2 & 420 \\
\hline 2 & 210 \\
\hline 3 & 105 \\
\hline 5 & 35 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 420=2 \times 2 \times 3 \times 5 \times 7 \\
& =2^2 \times 3 \times 5 \times 7
\end{aligned}
$
Question 9
Ans.
We have
$\begin{array}{r|r}
7 & 637 \\
\hline 7 & 91 \\
\hline 13 & 13 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 637=7 \times 7 \times 13 \\
& =7^2 \times 13
\end{aligned}
$
Question 10
945Ans.
We have
$\begin{array}{r|r}
3 & 945 \\
\hline 3 & 315 \\
\hline 3 & 105 \\
\hline 5 & 35 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 945=3 \times 3 \times 3 \times 5 \times 7 \\
& =3^3 \times 5 \times 7
\end{aligned}
$
$\begin{array}{r|r}
7 & 637 \\
\hline 7 & 91 \\
\hline 13 & 13 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 637=7 \times 7 \times 13 \\
& =7^2 \times 13
\end{aligned}
$
Question 10
945Ans.
We have
$\begin{array}{r|r}
3 & 945 \\
\hline 3 & 315 \\
\hline 3 & 105 \\
\hline 5 & 35 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 945=3 \times 3 \times 3 \times 5 \times 7 \\
& =3^3 \times 5 \times 7
\end{aligned}
$
Question 11
1224
Ans.
We have
$\begin{array}{r|r}
2 & 1224 \\
\hline 2 & 612 \\
\hline 2 & 306 \\
\hline 3 & 153 \\
\hline 3 & 51 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1224=2 \times 2 \times 2 \times 3 \times 3 \times 17 \\
& =2^3 \times 3^2 \times 17
\end{aligned}
$
Question 12
1323
Ans.
$\begin{array}{r|r}
2 & 1224 \\
\hline 2 & 612 \\
\hline 2 & 306 \\
\hline 3 & 153 \\
\hline 3 & 51 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1224=2 \times 2 \times 2 \times 3 \times 3 \times 17 \\
& =2^3 \times 3^2 \times 17
\end{aligned}
$
Question 12
Ans.
We have
$\begin{array}{l|r}
3 & 1323 \\
\hline 3 & 441 \\
\hline 3 & 147 \\
\hline 7 & 49 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1323=3 \times 3 \times 3 \times 7 \times 7 \\
& =3^3 \times 7^2
\end{aligned}
$
Question 13
8712
Ans.
Question 17
4641
Ans.
$\begin{array}{l|r}
3 & 1323 \\
\hline 3 & 441 \\
\hline 3 & 147 \\
\hline 7 & 49 \\
\hline 7 & 7 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1323=3 \times 3 \times 3 \times 7 \times 7 \\
& =3^3 \times 7^2
\end{aligned}
$
Question 13
Ans.
We have
$\begin{array}{r|r}
2 & 8712 \\
\hline 2 & 4356 \\
\hline 2 & 2178 \\
\hline 3 & 1089 \\
\hline 3 & 363 \\
\hline 11 & 121 \\
\hline 11 & 11 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 8712=2 \times 2 \times 2 \times 3 \times 3 \times 11 \times 11 \\
& =2^3 \times 3^2 \times 11^2 .
\end{aligned}
$
Question 14
$\begin{array}{r|r}
2 & 8712 \\
\hline 2 & 4356 \\
\hline 2 & 2178 \\
\hline 3 & 1089 \\
\hline 3 & 363 \\
\hline 11 & 121 \\
\hline 11 & 11 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 8712=2 \times 2 \times 2 \times 3 \times 3 \times 11 \times 11 \\
& =2^3 \times 3^2 \times 11^2 .
\end{aligned}
$
Question 14
9317
Ans. We have
$\begin{array}{r|r}
7 & 9317 \\
\hline 11 & 1331 \\
\hline 11 & 121 \\
\hline 11 & 11 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 9317=7 \times 11 \times 11 \times 11 \\
& =7 \times 11^3
\end{aligned}
$
Question 15
1035
Ans.
We have
$\begin{array}{r|r}
3 & 1035 \\
\hline 3 & 345 \\
\hline 5 & 115 \\
\hline 23 & 23 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1035=3 \times 3 \times 5 \times 23 \\
& =3^2 \times 5 \times 23
\end{aligned}
$
Question 16
1197
Ans.
$\begin{array}{r|r}
7 & 9317 \\
\hline 11 & 1331 \\
\hline 11 & 121 \\
\hline 11 & 11 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 9317=7 \times 11 \times 11 \times 11 \\
& =7 \times 11^3
\end{aligned}
$
Question 15
Ans.
We have
$\begin{array}{r|r}
3 & 1035 \\
\hline 3 & 345 \\
\hline 5 & 115 \\
\hline 23 & 23 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1035=3 \times 3 \times 5 \times 23 \\
& =3^2 \times 5 \times 23
\end{aligned}
$
Question 16
Ans.
We have
$\begin{array}{r|r}
3 & 1197 \\
\hline 3 & 399 \\
\hline 7 & 133 \\
\hline 19 & 19 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1197=3 \times 3 \times 7 \times 19 \\
& =3^2 \times 7 \times 19
\end{aligned}
$
$\begin{array}{r|r}
3 & 1197 \\
\hline 3 & 399 \\
\hline 7 & 133 \\
\hline 19 & 19 \\
\hline & 1
\end{array}$
$
\begin{aligned}
& \therefore 1197=3 \times 3 \times 7 \times 19 \\
& =3^2 \times 7 \times 19
\end{aligned}
$
Question 17
4641
Ans.
We have
$\begin{array}{r|r}
3 & 4641 \\
\hline 7 & 1547 \\
\hline 13 & 221 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$
\therefore 4641=3 \times 7 \times 13 \times 17 .
$
Question 18
4335
Ans.
$\begin{array}{r|r}
3 & 4641 \\
\hline 7 & 1547 \\
\hline 13 & 221 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$
\therefore 4641=3 \times 7 \times 13 \times 17 .
$
Question 18
4335
Ans.
We have
$\begin{array}{r|r}
3 & 4335 \\
\hline 5 & 1445 \\
\hline 17 & 289 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$\therefore 4335=3 \times 5 \times 17 \times 17=3 \times 5 \times 17^2$.
Question 19
2907
Ans.
$\begin{array}{r|r}
3 & 4335 \\
\hline 5 & 1445 \\
\hline 17 & 289 \\
\hline 17 & 17 \\
\hline & 1
\end{array}$
$\therefore 4335=3 \times 5 \times 17 \times 17=3 \times 5 \times 17^2$.
Question 19
2907
Ans.
We have
$\begin{array}{r|r}
3 & 2907 \\
\hline 3 & 969 \\
\hline 17 & 323 \\
\hline 19 & 19 \\
\hline & 1
\end{array}$
$
\therefore 2907=3 \times 3 \times 17 \times 19=3^2 \times 17 \times 19 .$
Question 20
$\begin{array}{r|r}
5 & 13915 \\
\hline 11 & 2783 \\
\hline 11 & 253 \\
\hline 23 & 23 \\
\hline & 1
\end{array}$
$
\therefore 13915=5 \times 11 \times 11 \times 23=5 \times 11^2 \times 23
$
$\begin{array}{r|r}
3 & 2907 \\
\hline 3 & 969 \\
\hline 17 & 323 \\
\hline 19 & 19 \\
\hline & 1
\end{array}$
$
\therefore 2907=3 \times 3 \times 17 \times 19=3^2 \times 17 \times 19 .$
Question 20
13915
Ans. We have$\begin{array}{r|r}
5 & 13915 \\
\hline 11 & 2783 \\
\hline 11 & 253 \\
\hline 23 & 23 \\
\hline & 1
\end{array}$
$
\therefore 13915=5 \times 11 \times 11 \times 23=5 \times 11^2 \times 23
$
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