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(i) We have,

Nội dung chính

    What smallest number should 7803 be multiplied with so that the product becomes a perfect cube ?`?What is the smallest number by which 6912 must be divided to make it a perfect cube?What is the smallest number by which 10976 must be divided to make it a perfect cube?What is the smallest number with which 7803 must be divided so that the quotient is a perfect cube?

1536 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3

After grouping the prime factors in triplets, it’s seen that one factor 3 is left without grouping.

1536 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × 3

So, in order to
make it a perfect cube, it must be divided by 3.

Thus, the smallest number by which 1536 must be divided to obtain a perfect cube is 3.

(ii) We have,

10985 = 5 × 13 × 13 × 13

After grouping the prime factors in triplet, it’s seen that
one factor 5 is left without grouping.

10985 = 5 × (13 × 13 × 13)

So, it must be divided by 5 in order to get a perfect cube.

Thus, the required smallest number is 5.

(iii) We have,

28672 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 ×
7

After grouping the prime factors in triplets, it’s seen that one factor 7 is left without grouping.

28672 = (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × (2 × 2 × 2) × 7

So, it must be divided by 7 in order to get a perfect cube.

Thus, the
required smallest number is 7.

(iv) 13718 = 2 × 19 × 19 × 19

After grouping the prime factors in triplets, it’s seen that one factor 2 is left without grouping.

13718 = 2 × (19 × 19 × 19)

So, it must be divided by 2 in order to get a
perfect cube.

Thus, the required smallest number is 2.

(i) 675 

Prime factors of 675 = 3 × 3 × 3 × 5 × 5 = 33 × 52 

We find that 675 is not a perfect cube. 

Hence, for making the quotient a perfect cube we divide it by 52 = 25, which gives 27 as quotient and we know that 27 is a perfect cube .

(ii) 8640 

Prime factors of 8640 = 2 × 2 × 2 × 2
× 2 × 2 × 3 × 3 × 3 × 5 = 23 × 23 × 33 × 5 

We find that 8640 is not a perfect cube. 

Hence, for making the quotient a perfect cube we divide it by 5 , which gives 1728 as quotient and we know that 1728 is a perfect cube.

(iii) 1600 

Prime factors of 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 = 23 × 23 × 52 

We
find that 1600 is not a perfect cube. 

Hence, for making 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

(iv) 8788 

Prime factors of 8788 = 2 × 2 × 13 × 13 × 13 = 22 × 133 

We find that 8788 is not a perfect cube. 

Hence, for making the quotient a perfect cube we
divide it by 4, which gives 2197 as quotient and we know that 2197 is a perfect cube

(v) 7803 

Prime factors of 7803 = 3 × 3 × 3 × 17 × 17 = 33 × 172 

We find that 7803 is not a perfect cube. 

Hence, for making 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

(vi)
107811 

Prime factors of 107811 = 3 × 3 × 3 × 3 × 11 × 11 × 11 = 33 × 113 × 3 

We find that 107811 is not a perfect cube. 

Hence, for making the quotient a perfect cube we divide it by 3, which gives 35937 as quotient and we know that 35937 is a perfect cube.

(vii) 35721 

Prime factors of 35721 = 3 × 3 × 3 × 3 × 3 × 3 × 7 × 7 =
33 × 33 × 72 

We find that 35721 is not a perfect cube. 

Hence, for making 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

(viii) 243000 

Prime factors of 243000 = 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 5 × 5 × 5 = 23 × 33 × 53 × 32 

We find that 243000 is not a perfect cube. 

Hence, for making 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

What smallest number should 7803 be multiplied with so that the product becomes a perfect cube ?`?

Hence, `17` is the smallest number by which `7803` should be multiplied to make it a perfect cube.

What is the smallest number by which 6912 must be divided to make it a perfect cube?

To do: To find the smallest number by which 6912 must be divided so that the number formed is a perfect cube. Therefore, we should divide 6912 by 22=4 2 2 = 4 , the smallest number to get 1728 which is a cube of 12 .

What is the smallest number by which 10976 must be divided to make it a perfect cube?

Hence, the smallest number by which 10976 must be multiplied to obtain a perfect cube is 2.

What is the smallest number with which 7803 must be divided so that the quotient is a perfect cube?

Hence, 7803 is not a perfect cube. We divide it by 172 = 289 to make the quotient a perfect cube, which gives 27 as quotient which is a perfect cube. Hence, 289 is the required smallest number.
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