Answer:
The smallest perfect square number which is exactly divisible by each of the 4, 6, 10 and 12 is 900 and [tex]\sqrt{900} [/tex] = 30
Step-by-step explanation:
The smallest perfect square number which is exactly divisible by each of the 4, 6, 10 and 12 is 7056
Step-by-step explanation:
We have to find the smallest perfect square number which is exactly divisible by each of the 4, 6, 10 and 12.
To find the smallest perfect square number which is exactly divisible by each of the 4, 6, 10 and 12, we have to first find the smallest number which is divisible by 4, 6, 10 and 12.
So, Smallest number divisible by 4, 6, 10 and 12 = LCM(4, 6, 10, 12).
[tex]\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:4 \: , \: 6\:, \: 10\:,\:12 \: \: }}}\\{\underline{\sf{2}}}&{\underline{\sf{\:\:2 \: , \: 3\:, \: 5\:,\:6 \: \: }}}\\ {\underline{\sf{3}}}&{\underline{\sf{\:\:1 \: , \: 3\:, \: 5\:,\:3 \: \: }}}\\{\underline{\sf{}}}&{\sf{\:1, \: \:1\:, \: 5 \:,\:1 \: }} \end{array}\end{gathered}\end{gathered}\end{gathered} \\ \\ \end{gathered} [/tex]
So,
[tex]\sf \:LCM(4, 6, 10, 12) = 2 \times 2 \times 3 \times 5 \\ \\ [/tex]
[tex]\sf\implies \sf \:LCM(4, 6, 10, 12) = 60 \\ \\ [/tex]
So, 60 is the smallest square number which is exactly divisible by each of the 4, 6, 10 and 12.
[tex]\sf \:Prime \: factorization \: of \: 60 = 2 \times 2 \times 3 \times 5 \\ \\[/tex]
We concluded from above factorization that prime factors 3 and 5 are not in pairs.
So, in order to get a perfect square, each factor of 60 must be in pairs.
[tex]\sf\implies 60 \: must \: be \: multiplied \: by \: 3\times 5 , \: i.e \: by \: 15 \\ \\[/tex]
[tex]\sf\implies Required\:number = 60 × 15 = 900 \\ \\[/tex]
Hence,
The smallest perfect square number which is exactly divisible by each of the 4, 6, 10 and 12 is 900
and
[tex]\sf \: \sqrt{900} = \sqrt{2.2.3.3.5.5} = 2.3.5 = 30 \\ \\ [/tex]