To do:
We have to find the smallest number by which the given numbers must be divided so that the quotient is a perfect cube.
Solution:
(i) Prime factorisation of $675=3\times3\times3\times5\times5$
Grouping the factors in triplets of equal factors, we find that $5\times5$ is not a complete triplet.
Therefore, dividing $675$ by $5\times5=25$, we get,
$675\div25=3\times3\times3\times5\times5\div25$
$=3\times3\times3$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 25.
(ii) Prime factorisation of $8640=2\times2\times2\times2\times2\times2\times3\times3\times3\times5$
Grouping the factors in triplets of equal factors, we find that $5$ is not a complete triplet.
Therefore, dividing $8640$ by $5$, we get,
$8640\div5=2\times2\times2\times2\times2\times2\times3\times3\times3\times5\div5$
$=2\times2\times2\times2\times2\times2\times3\times3\times3$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 5.
(iii) Prime factorisation of $1600=2\times2\times2\times2\times2\times2\times5\times5$
Grouping the factors in triplets of equal factors, we find that $5\times5$ is not a complete triplet.
Therefore, dividing $1600$ by $5\times5=25$, we get,
$1600\div25=2\times2\times2\times2\times2\times2\times5\times5\div25$
$=2\times2\times2\times2\times2\times2$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 25.
(iv) Prime factorisation of $8788=2\times2\times13\times13\times13$
Grouping the factors in triplets of equal factors, we find that $2\times2$ is not a complete triplet.
Therefore, dividing $8788$ by $2\times2=4$, we get,
$8788\div4=2\times2\times13\times13\times13\div4$
$=13\times13\times13$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 4.
(v) Prime factorisation of $7803=3\times3\times3\times17\times17$
Grouping the factors in triplets of equal factors, we find that $17\times17$ is not a complete triplet.
Therefore, dividing $7803$ by $17\times17=289$, we get,
$7803\div289=3\times3\times3\times17\times17\div289$
$=3\times3\times3$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 289.
(vi) Prime factorisation of $107811=3\times3\times3\times3\times11\times11\times11$
Grouping the factors in triplets of equal factors, we find that $3$ is not a complete triplet.
Therefore, dividing $107811$ by $3$, we get,
$107811\div3=3\times3\times3\times3\times11\times11\times11\div3$
$=3\times3\times3\times11\times11\times11$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 3.
(vii) Prime factorisation of $35721=3\times3\times3\times3\times3\times3\times7\times7$
Grouping the factors in triplets of equal factors, we find that $7\times7$ is not a complete triplet.
Therefore, dividing $35721$ by $7\times7=49$, we get,
$35721\div49=3\times3\times3\times3\times3\times3\times7\times7\div49$
$=3\times3\times3\times3\times3\times3$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 49.
(viii) Prime factorisation of $243000=2\times2\times2\times3\times3\times3\times3\times3\times5\times5\times5$
Grouping the factors in triplets of equal factors, we find that $3\times3$ is not a complete triplet.
Therefore, dividing $243000$ by $3\times3=9$, we get,
$243000\div9=2\times2\times2\times3\times3\times3\times3\times3\times5\times5\times5\div9$
$=2\times2\times2\times3\times3\times3\times5\times5\times5$
Hence, the smallest number by which the given number must be divided so that the quotient is a perfect cube is 9.
Text Solution
Solution : By using prime factorization we could find:<br>(i)`675`<br>`675=3^3times5^2`<br>As we can see that `5^2` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `5^2=25`, which gives `27` as quotient where, `27` is a perfect cube. <br><br>(ii)`8640`<br>`8640=2^6times3^3times5`<br>As we can see that `5` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `5`, which gives `1728` as quotient where, `1728` is a perfect cube.<br><br>(iii)`1600`<br>`1600=2^6times5^2`<br>As we can see that `5^2` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `5^2=25`, which gives `64` as quotient where, `64` is a perfect cube.<br><br>(iv)`8788`<br>`8788=2^2times13^3`<br>As we can see that `2^2` is not cubed.<br>Hence, to make the quotient a perfect cube we divide it by `2^2=4`, which gives `2197` as quotient where, `2197` is a perfect cube.
(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.
We have to find the smallest number by which 675 must be multiplied to obtain a perfect cube Solution 675 On prime factorisation of 675 we get,
675 = 3×3×3×5×5
By assorting the factors in triplets of equal factors, 675 = (3×3×3) x 5×5
Here, 5 cannot be arranged into triplets of equal factors.
∴ We will multiply 675 by 5 to get perfect cube.
Answer
The smallest number by which 675 must be multiplied to obtain a perfect cube is 5.
Was this answer helpful?
4 (23)
Thank you. Your Feedback will Help us Serve you better.