Question

Find the smallest number by which 2916 should be divided so that the quotient is a perfect cube. Also find the cube root of the quotient

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Answer 1

$\Large {\underline { \sf \orange{Clarification :}}}$

Here, as per the given question we have to find the smallest number by which 2916 should be divided so that the quotient is a perfect cube. And we also have to find the cube root of the quotient.

In order to tackle this problem, we'll first resolve the given number into prime factors. After that, we'll make the group of 3 common factors and the number which can't be formed as the group of 3 common factors is the number that should be divided so that the quotient is a perfect cube.

Then, we'll find the cube root of the quotient by prime factorization method.

$\bf \red { \dag }$ How to find cube root by prime factorization ?

$\longmapsto$ In prime factorization method, firstly we find the prime factors of the given number by prime factorization. Then, we form groups of triplets of like factors and from each group of triplets we pick out one prime factor and then we multiply the the factors so picked , the product we get is the cube root of the given number.

$\Large {\underline { \sf \orange{Explication \: of \: Steps :}}}$

Resolving the number into prime factors,

$\begin{array}{c | c} \underline{\sf{2}}&\underline{\sf{2916}} \\ \underline{\sf{2}}&\underline{\sf{1458}} \\\underline{\sf{3}}&\underline{\sf{ \: 729 \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: 243 \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: \: 81 \: \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: \: 27 \: \: }} \\\underline{\sf{ \: 3 \: }}&\underline{\sf{ \: \: \: 9 \: \: \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: \: \: 3 \: \: \: }} \\ \: &1\end{array}$

We get that,

$\longrightarrow$ 2916 = 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3

• Making the group of triplets of common factors.

$\longrightarrow$ (2 × 2) can't be formed as the group of triplets. So, (2×2) i.e, 4 is the smallest number by which 2916 should be divided so that the quotient is a perfect cube.

Finding Quotient,

$\longrightarrow$ Quotient = Given Number ÷ 4

$\longrightarrow \sf { Quotient = \dfrac{2916}{4} }$

$\longrightarrow \sf { Quotient = 729 }$ ★

Calculating the cube root of 729 :

By prime factorization,

$\begin{array}{c | c} \underline{ \sf3}&\underline{\sf{ \: 729 \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: 243 \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: \: 81 \: \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: \: 27 \: \: }} \\\underline{\sf{ \: 3 \: }}&\underline{\sf{ \: \: \: 9 \: \: \: }} \\ \underline{\sf{3}}&\underline{\sf{ \: \: \: 3 \: \: \: }} \\ \: &1\end{array}$

$\longrightarrow$ 729 = 3 × 3 × 3 × 3 × 3 × 3

• Pick out one prime factor from each group of triplets of common factors.

$\longrightarrow \sf { \sqrt[3]{729} = 3 \times 3}$

$\longrightarrow$ ❝ $\boxed{ \sf \orange { \sqrt[3]{729} = 9}}$ ❞

❝ Therefore , 4 is the smallest number by which 2916 should be divided so that the quotient is a perfect cube and the the cube root of the quotient is 9.❞

Answer 2

Step-by-step explanation:

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