find the smallest perfect square number which is exactly divisible by each of the 6,7,8
Answers 2
Answer:
7056
Step-by-step explanation:
LCM OF 6, 7 and 8 = 336
Therefore, 336 = 2 * 2 * 2 * 2 * 3 * 7
Therefore, Perfect square = 2 * 2 * 2 * 2 * 3 * 3 * 7 * 7
= 7056
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Author:
jacquelynfitzpatrick
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Answer:
The smallest perfect square number which is exactly divisible by each of the 6, 7, 8 is 7056
Step-by-step explanation:
We have to find the smallest perfect square number which is exactly divisible by each of the 6, 7, 8.
To find the smallest perfect square number which is exactly divisible by each of the 6,7,8, we have to first find the smallest number which is divisible by 6, 7, 8.
So, Smallest number divisible by 6, 7, 8 = LCM(6, 7, 8).
[tex]\begin{gathered}\begin{gathered}\begin{gathered} \:\: \begin{array}{c|c} {\underline{\sf{2}}}&{\underline{\sf{\:\:6\:, \: 7\:,\:8 \: \: }}}\\ {\underline{\sf{}}}&{\sf{\:\:3\:, \: 7 \:,\:4}} \end{array}\end{gathered}\end{gathered}\end{gathered} \\ \\ [/tex]
So,
[tex]\sf \:LCM(6, 7, 8) = 2 \times 3 \times 7 \times 4 \\ \\ [/tex]
[tex]\sf \:LCM(6, 7, 8) = 168 \\ \\ [/tex]
So, 168 is the smallest square number which is exactly divisible by each of the 6,7,8.
[tex]\sf \:Prime \: factorization \: of \: 168 = 2 \times 2 \times 2 \times 3 \times 7 \\ \\ [/tex]
We concluded from above factorization that prime factors 2, 3 and 7 are not in pairs.
So, in order to get a perfect square, each factor of 168 must be in pairs.
[tex]\sf\implies 168 \: must \: be \: multiplied \: by \: 2 \times 3 \times 7, \: i.e \: by \: 42 \\ \\ [/tex]
[tex]\sf\implies Required\:number = 168 × 42 = 7056 \\ \\ [/tex]
Hence,
The smallest perfect square number which is exactly divisible by each of the 6,7,8 is 7056
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Author:
nuriaauyd
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