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41. Find L.C.M. and H.C.F. of 2 x 2!, 4! and 3 x 5!

2 cross times 2 factorial space equals space 2 cross times 2 cross times 1 space equals space 2 squared space space space

space space space 4 factorial space equals space 4 cross times 3 cross times 2 cross times 1 space equals space 2 cubed cross times 3 to the power of 1

3 cross times 5 factorial space equals space 3 cross times 5 cross times 4 cross times 3 cross times 2 cross times 1 space equals space 2 cubed cross times 3 squared cross times 5 to the power of 1

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L.C.M. =

space space space 2 cubed cross times 3 squared cross times 5 to the power of 1 equals 2 cross times 2 cross times 2 cross times 3 cross times 3 cross times 5

= (5 x 4 x 3 x 2 x 1) x 3 = 3 x 5!

and H.C.F. = 2² = 2 x (2 x 1) = 2 x 2!

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$space space space space space space space fraction numerator left parenthesis 2 straight n right parenthesis factorial over denominator straight n factorial end fraction space equals space left square bracket 1 times 3 times 5 times........... space left parenthesis 2 straight n minus 1 right parenthesis right square bracket 2 to the power of straight n$