question_answer 1)

            Find the common factors of the given terms: (i) \[12x,\,36\]                   (ii) \[2y,\,22xy\]                                (iii) \[14pq,\,28{{p}^{2}}{{q}^{2}}\]           (iv) \[2x,\,3{{x}^{2}},4\] (v) \[6abc,\,24a{{b}^{2}},\,12{{a}^{2}}b\]              (vi) \[16{{x}^{3}},\,-4{{x}^{2}},\,32x\]     (vii) \[10pq,\,20qr,\,30rp\]   (viii) \[3{{x}^{2}}{{y}^{3}},\,10{{x}^{3}}{{y}^{2}},\,6{{x}^{2}}{{y}^{2}}z\].

Answer:

                (i) \[12x,\,36\]                   \[12x=\underline{2}\times \underline{\underline{2}}\times \underset{\centerdot }{\mathop{3}}\,\times x\] \[36=\underline{2}\times \underline{\underline{2}}\times \underset{\centerdot }{\mathop{3}}\,\times 3\] Common prime factors are 2 (occurs twice) and 3. \[\therefore \]  H.C.F. \[=2\times 2\times 3=12\] (ii) \[2y,\,22xy\] \[2y=\underline{2}\times \underline{\underline{y}}\] \[22xy=\underline{2}\times 11\times x\times \underline{\underline{y}}\]                 Common factors are 2 and \[y\]. \[\therefore \]  H.C.F.    \[=2\times y=2y\] (iii) \[14pq,\,28{{p}^{2}}{{q}^{2}}\]           \[14pq=\underline{2}\times \underline{\underline{7}}\times \underset{0}{\mathop{p}}\,\times \underset{w}{\mathop{q}}\,\] \[28{{p}^{2}}{{q}^{2}}=\underline{2}\times 2\times \underline{\underline{7}}\times \underset{0}{\mathop{p}}\,\times p\times \underset{w}{\mathop{q}}\,\times q\] Common factors are 2, 7, p and q. \[\therefore \]  H.C.F.\[=2\times 7\times p\times q=14pq\] (iv) \[2x,\,3{{x}^{2}},4\] \[2x=\underline{1}\times 2\times x\] \[3{{x}^{2}}=\underline{1}\times 3\times x\times x\] \[4=\underline{1}\times 2\times 2\]                 Common factor is 1 \[\therefore \]  H.C.F. = 1 (v) \[6abc,\,24a{{b}^{2}},\,12{{a}^{2}}b\] \[6abc=\underline{2}\times \underline{\underline{3}}\times \underset{0}{\mathop{a}}\,\times \underset{w}{\mathop{b}}\,\times c\] \[24a{{b}^{2}}=\underline{2}\times 2\times 2\times \underline{\underline{3}}\times \underset{0}{\mathop{a}}\,\times \underset{w}{\mathop{b}}\,\times b\] \[12{{a}^{2}}b=\underline{2}\times 2\times \underline{\underline{3}}\times \underset{0}{\mathop{a}}\,\times a\times \underset{w}{\mathop{b}}\,\] Common factors are 2, 3 a and b \[\therefore \]  H.C.F. \[=2\times 3\times a\times b\] \[=6ab\] (vi) \[16{{x}^{3}},\,-4{{x}^{2}},\,32x\] \[16{{x}^{3}}=\underline{2}\times \underline{\underline{2}}\times 2\times 2\times \underset{0}{\mathop{x}}\,\times x\times x\] \[-4{{x}^{2}}=-1\times \underline{2}\times \underline{\underline{2}}\times \underset{0}{\mathop{x}}\,\times x\] \[32x=\underline{2}\times \underline{\underline{2}}\times 2\times 2\times \underset{0}{\mathop{x}}\,\] Common factors are 2 (occurs twice and \[x\](occurs once). \[\therefore \]  H.C.F. \[=2\times 2\times x=4x\] (vii) \[10pq,\,20qr,\,30rp\] \[10pq=\underline{2}\times \underline{\underline{5}}\times p\times q\] \[20qr=\underline{2}\times 2\times \underline{\underline{5}}\times q\times r\] \[30rp=\underline{2}\times 3\times \underline{\underline{5}}\times r\times p\] Common factors are 2 and 5. \[\therefore \]  H.C.F. \[2\times 5=10\] (viii) \[3{{x}^{2}}{{y}^{3}},\,10{{x}^{3}}{{y}^{2}},\,6{{x}^{2}}{{y}^{2}}z\]. \[3{{x}^{2}}{{y}^{3}}=3\times \underline{x}\times \underline{\underline{x}}\times \underset{0}{\mathop{y}}\,\times \underset{00}{\mathop{y}}\,\times y\] \[10{{x}^{3}}{{y}^{2}}=2\times 5\times \underline{x}\times \underline{\underline{x}}\times x\times \underset{0}{\mathop{y}}\,\times \underset{00}{\mathop{y}}\,\] \[6{{x}^{2}}{{y}^{2}}z=2\times 3\times \underline{x}\times \underline{\underline{x}}\times \underset{0}{\mathop{y}}\,\times \underset{00}{\mathop{y}}\,\times z\] Common factors are \[x\] (occurs twice) and \[y\](occurs twice) \[\therefore \]  H.C.F. \[=x\times x\times y\times y={{x}^{2}}{{y}^{2}}\]