\begin{array}{l}
\text {Q1. In which of the following expressions, prime factorization has been done? } \\
\text {(i) } \quad 24=2 \times 3 \times 4 \\
\text {(ii) } \quad 56=1 \times 7 \times 2 \times 2 \times 2 \\
\text {(iii) } \quad 70=2 \times 5 \times 7 \\
\text {(iv) } \quad 54=2 \times 3 \times 9 \\ \\\text {Sol. (i) } \quad 24=2 \times 3 \times 4 \\
\text { is not a prime factorization as 4 is not a prime number.} \\ \\\text {(ii) } \quad 56=1 \times 7 \times 2 \times 2 \times 2 \\
\text {is not a prime factorization as 1 is not a prime number. } \\ \\\text {(iii) } \quad 70=2 \times 5 \times 7 \\
\text {is a prime factorization as 2,5 and 7 are prime numbers.} \\ \\\text {(iv) } \quad 54=2 \times 3 \times 9 \\
\text {is not a prime factorization as 9 is not a prime number.} \\ \\\text {Q2. Determine prime factorization of each of the following numbers: } \\
\text {(i) 216 } \quad
\text {(ii) 420 } \\
\text {(iii) 468 } \quad
\text {(iv) 945 } \\
\text {(v) 7325 } \quad
\text {(vi) 13915 } \\ \\\text {Sol. (i)} \\
\end{array}\begin{array}{l|l}
2 & 216 \\
\hline
2 & 108 \\
\hline
2 & 54 \\
\hline
3 & 27 \\
\hline
3 & 9 \\
\hline
3 & 3 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 216 is } \\
2 \times 2 \times 2 \times 3 \times 3 \times 3 \\ \\
\end{array}\begin{array}{l}
\text {(ii) 420 } \\
\end{array}\begin{array}{l|l}
2 & 420 \\
\hline
2 & 210 \\
\hline
3 & 105 \\
\hline
5 & 35 \\
\hline
7 & 7 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 420 is } \\
2 \times 2 \times 3 \times 5 \times 7 \\
\end{array}\begin{array}{l}
\text {(iii) 468 } \\
\end{array}\begin{array}{l|l}
2 & 468 \\
\hline
2 & 234 \\
\hline
3 & 117 \\
\hline
3 & 39 \\
\hline
13 & 13 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 468 is } \\
2 \times 2 \times 3 \times 3 \times 13 \\
\end{array}\begin{array}{l}
\text {(iv) 945 } \\
\end{array}\begin{array}{l|l}
3 & 945 \\
\hline
3 & 315 \\
\hline
3 & 105 \\
\hline
5 & 35 \\
\hline
7 & 7 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 945 is } \\
3 \times 3 \times 3 \times 5 \times 7 \\
\end{array}\begin{array}{l}
\text {(v) 7325 } \\
\end{array}\begin{array}{l|l}
5 & 7325 \\
\hline
5 & 1465 \\
\hline
293 & 293 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 7325 is } \\
5 \times 5 \times 293 \\
\end{array}\begin{array}{l}
\text {(vi) 13915 } \\
\end{array}\begin{array}{l|l}
5 & 13915 \\
\hline
11 & 2783 \\
\hline
11 & 253 \\
\hline
23 & 23 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 13915 is } \\
5 \times 11 \times 11 \times 23 \\
\end{array}\begin{array}{l}
\text {Q3. Write the smallest 4-digit number and express it as a product of primes. } \\ \\
\text {Sol. 1000 is the smallest 4-digit number.} \\
1000 = 2 \times 500 \\
= 2 \times 2 \times 250 \\
= 2 \times 2 \times 2 \times 125 \\
= 2 \times 2 \times 2 \times 5 \times 25 \\
= 2 \times 2 \times 2 \times 5 \times 5 \times 5 \\
\therefore \quad 1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5 \\ \\
\end{array}\begin{array}{l}
\text {Q4. Write the largest 4-digit number and give its prime factorization. } \\
\text {Sol. The largest 4-digit number is 9999 } \\
\end{array}\begin{array}{l|l}
3 & 9999 \\
\hline
3 & 3333 \\
\hline
11 & 1111 \\
\hline
101 & 101 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 9999 is } \\
3 \times 3 \times 11 \times 101 \\
\end{array}\begin{array}{l}
\text {Q5. Find the prime factors of 1729. Arrange the factors in ascending order, } \\
\text {and find the relation between two consecutive prime factors.} \\
\end{array}\begin{array}{l}
\text {Sol. 1729 } \\
\end{array}\begin{array}{l|l}
7 & 1729 \\
\hline
13 & 247 \\
\hline
19 & 19 \\
\hline
& 1 \\
\end{array}\begin{array}{l}
\text {Hence, the prime factorization of 1729 is } \\
7 \times 13\times 19 \\
\text {The consecutive prime factors of 1729 are 7,13 and 19.} \\
\text {Clearly, } 13-7=6 \text { and } 19-13=6 \\
\text {Here, the latter prime factor is 6 more than the previous one, in the set of prime factors.}
\end{array}
\begin{array}{l}
\text {Q6. Which factors are not included in the prime factorization of a composite number?} \\ \\
\text {Sol. 1 and the number itself are not included in the prime factorization of a composite number.} \\
\text {Q7. Here are two different factor trees for 60. Write the missing numbers:} \\ \\
\end{array}
\begin{array}{l}
\text {Sol. (i) Since } 6 = 2 \times 3 \text { and } 10 = 5 \times 2 \\
\text {We have } \\
\end{array}
\begin{array}{l}
\text {(ii) We know } 60 = 30 \times 2 \\
30 = 10 \times 3 \\
10 = 2 \times 5 \\
\text {We have } \\
\end{array}
