प्रश्न 1:– निम्नलिखित संख्याओं को अभाज्य गुणनखंडों के गुणनफल के रूप में व्यक्त कीजिए ।
(i) 140
\[\begin{array}{c|c}2 & 140 \\\hline2 & 70 \\\hline5 & 35 \\\hline7 & 7 \\\hline & 1 \\\hline\end{array}\]\[140 = 2 \times 2 \times 5 \times 7\]\[ = 2^2 \times 5 \times 7\](ii) 156\[\begin{array}{c|c}2 & 156 \\\hline2 & 78 \\\hline3 & 39 \\\hline13 & 13 \\\hline & 1 \\\hline\end{array}\]\[156 = 2 \times 2 \times 3 \times 13\]\[ = 2^2 \times 3 \times 13\](iii) 3825\[\begin{array}{c|c}3 & 3825 \\\hline3 & 1275 \\\hline5 & 425 \\\hline5 & 85 \\\hline17 & 17 \\\hline & 1 \\\hline\end{array}\]\[3825 = 3 \times 3 \times 5 \times 5 \times 17\]\[ = 3^2 \times 5^2 \times 17\](iv) 5005\[\begin{array}{c|c}5 & 5005 \\\hline7 & 1001 \\\hline11 & 143 \\\hline13 & 13 \\\hline & 1 \\\hline\end{array}\]\[5005 = 5 \times 7 \times 11 \times 13\](v) 7429\[\begin{array}{c|c}17 & 7429 \\\hline19 & 437 \\\hline23 & 23 \\\hline & 1 \\\hline\end{array}\]\[7429 = 17 \times 19 \times 23\]
प्रश्न 2:– निम्नलिखित संख्याओं का अभाज्य गुणनखंड विधि द्वारा HCF एवं LCM ज्ञात की
दो संख्याओं का गुणनफल = HCF × LCM
((i) 26 और 91
26 के लिए:\[\begin{array}{c|c}2 & 26 \\\hline13 & 13 \\\hline & 1 \\\hline\end{array}\]\[26 = 2 \times 13\]91 के लिए:\[\begin{array}{c|c}7 & 91 \\\hline13 & 13 \\\hline & 1 \\\hline\end{array}\]\[91 = 7 \times 13\]\[HCF = 13 \\ LCM = 2 \times 7 \times 13 = 182\]
जांच:\[26 \times 91 = 2366 \\13 \times 182 = 2366\]सही है।
(ii) 510 और 92
510 के लिए:\[\begin{array}{c|c}2 & 510 \\\hline3 & 255 \\\hline5 & 85 \\\hline17 & 17 \\\hline & 1 \\\hline\end{array}\]\[510 = 2 \times 3 \times 5 \times 17\] 92 के लिए:\[\begin{array}{c|c}2 & 92 \\\hline2 & 46 \\\hline23 & 23 \\\hline & 1 \\\hline\end{array}\]\[92 = 2 \times 2 \times 23\] \[HCF = 2 \\ LCM = 2 \times 3 \times 5 \times 17 \times 2 \times 23 = 23460\] जांच:\[510 \times 92 = 46920 \\2 \times 23460 = 46920\]सही है।
(iii) 336 और 54
336 के लिए:\[\begin{array}{c|c}2 & 336 \\\hline2 & 168 \\\hline2 & 84 \\\hline2 & 42 \\\hline3 & 21 \\\hline7 & 7 \\\hline & 1 \\\hline\end{array}\]\[336 = 2^4 \times 3 \times 7\] 54 के लिए:\[\begin{array}{c|c}2 & 54 \\\hline3 & 27 \\\hline3 & 9 \\\hline3 & 3 \\\hline & 1 \\\hline\end{array}\]\[54 = 2 \times 3^3\] \[HCF = 2 \times 3 = 6\\ LCM = 2^4 \times 3^3 \times 7 = 3024\] जांच:\[336 \times 54 = 18144 \\6 \times 3024 = 18144\]सही है।
प्रश्न 3:–निम्नलिखित संख्याओं के अभाज्य गुणनखंड विधि द्वारा HCF एवं LCM ज्ञात करे।
(i) 12, 15, 21\[\begin{array}{c|c}2 & 12 \\\hline2 & 6 \\\hline3 & 3 \\\hline & 1 \\\hline\end{array}\quad\begin{array}{c|c}3 & 15 \\\hline5 & 5 \\\hline & 1 \\\hline\end{array}\quad\begin{array}{c|c}3 & 21 \\\hline7 & 7 \\\hline & 1 \\\hline\end{array}\]\[12 = 2 \times 2 \times 3 \\15 = 3 \times 5 \\21 = 3 \times 7\]\[\text{HCF} = 3 \\\text{LCM} = 2 \times 2 \times 3 \times 5 \times 7 = 420\](ii) 17, 23, 29\[17 = 17 \\23 = 23 \\29 = 29\]\[\text{HCF} = 1 \\\text{LCM} = 17 \times 23 \times 29 = 11339\](iii) 8, 9, 25\[\begin{array}{c|c}2 & 8 \\\hline2 & 4 \\\hline2 & 2 \\\hline & 1 \\\hline\end{array}\quad\begin{array}{c|c}3 & 9 \\\hline3 & 3 \\\hline & 1 \\\hline\end{array}\quad\begin{array}{c|c}5 & 25 \\\hline5 & 5 \\\hline & 1 \\\hline\end{array}\]\[8 = 2 \times 2 \times 2 \\9 = 3 \times 3 \\25 = 5 \times 5\]\[\text{HCF} = 1 \\\text{LCM} = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 = 1800\]