What is the smallest number by which 1600 must be divided so that the question is a perfect square?

(i)First find the prime factors of 675

675 = 3 × 3 × 3 × 5 × 5

= 33 × 52

Since 675 is not a perfect cube.

To make the quotient a perfect cube we divide it by 52 = 25, which gives 27 as quotient

where, 27 is a perfect cube.

∴ 25 is the required smallest number.

(ii) 8640

First find the prime factors of 8640

8640 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5

= 23 × 23 × 33 × 5

Since 8640 is not a perfect cube.

To make the quotient a perfect cube we divide it by 5, which gives 1728 as quotient and

we know that 1728 is a perfect cube.

∴5 is the required smallest number.

(iii) 1600

First find the prime factors of 1600

1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5

= 23 × 23 × 52

Since 1600 is not a perfect cube.

To make the quotient a perfect cube we divide it by 52 = 25, which gives 64 as quotient

and we know that 64 is a perfect cube

∴ 25 is the required smallest number.

(iv) 8788

First find the prime factors of 8788

8788 = 2 × 2 × 13 × 13 × 13

= 22 × 133

Since 8788 is not a perfect cube.

To make the quotient a perfect cube we divide it by 4, which gives 2197 as quotient and

we know that 2197 is a perfect cube

∴ 4 is the required smallest number.

(v) 7803

First find the prime factors of 7803

7803 = 3 × 3 × 3 × 17 × 17

= 33 × 172

Since 7803 is not a perfect cube.

To make the quotient a perfect cube we divide it by 172 = 289, which gives 27 as quotient and we know that 27 is a perfect cube ∴ 289 is the required smallest number. (vi) 107811 First find the prime factors of 107811 107811 = 3 × 3 × 3 × 3 × 11 × 11 × 11 = 33 × 113 × 3 Since 107811 is not a perfect cube. To make the quotient a perfect cube we divide it by 3, which gives 35937 as quotient and we know that 35937 is a perfect cube. ∴ 3 is the required smallest number. (vii) 35721 First find the prime factors of 35721 35721 = 3 × 3 × 3 × 3 × 3 × 3 × 7 × 7 = 33 × 33 × 72 Since 35721 is not a perfect cube. To make the quotient a perfect cube we divide it by 72 = 49, which gives 729 as quotient and we know that 729 is a perfect cube ∴ 49 is the required smallest number. (viii) 243000 First find the prime factors of 243000 243000 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 = 23 × 33 × 53 × 32 Since 243000 is not a perfect cube. To make the quotient a perfect cube we divide it by 32 = 9, which gives 27000 as quotient and we know that 27000 is a perfect cube ∴ 9 is the required smallest number.

Answer

Verified

Hint: To solve this question, we will start with factorising the given number \[1600,\] where we will get some factors of the number, then we will make group of three (because as it is given that it needs to be perfect cube), so, after arranging the factors into group of three, the remaining number will be our required answer.

Complete step-by-step answer:

We have been given a number, i.e., \[1600,\] we need to find the smallest number by which \[1600\] must be divided so that the quotient is a perfect cube.The given number is \[1600,\] so we will start with factorizing the number \[1600.\]On factorization the number, we get\[1600{\text{ }} = {\text{ }}2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5\]So, we get the factors of \[1600,\] now we need to group the numbers in a group of three since it is given in the question that it has to be a perfect cube. We can see above that the number \[5\] does not form a triplet, because it only contains two \[5\prime s.\]Hence, the number, \[5 \times 5 = 25,\] i.e., \[25\] has to be divided so that the quotient becomes a perfect cube.Thus, the smallest number by which \[1600\] must be divided so that the quotient is a perfect cube is \[25.\]

Note: In the question, we were asked about the perfect cube, so, a perfect cube is an integer that is equal to some other integer which is raised to the third power. We refer to the perfect cube by raising a number to the third power as cubing the number.

For example, \[8\] is a perfect cube because of \[2,\] because, \[2 \times 2 \times 2 = 8.\]

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