Question

4. Find the smallest number by which 3645 must be divided so that it becomes a
perfect square. Also, find the square root of the resulting number
​

Answer 1

Given number :

• 3645

To find :

• The smallest number by which 3645 must be divided so that it becomes a perfect square.

• The square root of the resulting number.

Calculation :

Let us first resolve the number into prime factors by prime factorization :

$\large {\begin{array}{c|c} \underline{5} & \underline{3645} \\ \underline{3} & \underline{729} \\\underline{3} & \underline{243} \\ \underline{3} & \underline{81} \\ \underline{3} & \underline{9} \\ \underline{3} & \underline{3} \\ \: & 1 \end{array} }$

We get that :

→ 3645 = 5 × 3 × 3 × 3 × 3

Grouping into pairs.

→ 3645 = 5 × 3 × 3 × 3 × 3

→ 3645 = 5 × 3² × 3²

We get that,

• 5 left unpaired. So, 5 is the smallest number by which 3645 must be divided so that it becomes a perfect square.

Also,

→ New number = 5 × 5 × 3 × 3 × 3 × 3

→ New number = 5² × 3² × 3²

→ New number = ( 5 × 3 × 3 )²

→ New number = (45)² ⇒ 2025

Resulting number is 2025. Square root of the resulting number is 45.

Because, we got that :

→ (45)² ⇒ 2025

→ 45 ⇒ √2025

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